186:
261:
7068:
645:, it is a totally ordered set without any infinite decreasing sequence — though there may be infinite increasing sequences. Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labelled 0, the one after that 1, the next one 2, "and so on"), and to measure the "length" of the whole set by the least ordinal that is not a label for an element of the set. This "length" is called the
2780:
subclass (such as the class of all limit ordinals with countable cofinality). Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of a stationary class and a closed unbounded class is stationary. But the intersection of two stationary classes may be empty, e.g. the class of ordinals with cofinality ω with the class of ordinals with uncountable cofinality.
84:
6998:
43:
727:
552:, which measure the size of sets. Although the distinction between ordinals and cardinals is not always apparent on finite sets (one can go from one to the other just by counting labels), they are very different in the infinite case, where different infinite ordinals can correspond to sets having the same cardinal. Like other kinds of numbers, ordinals can be
4261:, "and so on", but all the subtlety lies in the "and so on"). One could try to do this systematically, but no matter what system is used to define and construct ordinals, there is always an ordinal that lies just above all the ordinals constructed by the system. Perhaps the most important ordinal that limits a system of construction in this manner is the
843:, which says, essentially, that any property that passes on from the predecessors of an element to that element itself must be true of all elements (of the given well-ordered set). If the states of a computation (computer program or game) can be well-ordered—in such a way that each step is followed by a "lower" step—then the computation will terminate.
2741:. For example, the class of all limit ordinals is closed and unbounded: this translates the fact that there is always a limit ordinal greater than a given ordinal, and that a limit of limit ordinals is a limit ordinal (a fortunate fact if the terminology is to make any sense at all!). The class of additively indecomposable ordinals, or the class of
1719:
Any nonzero ordinal has the minimum element, zero. It may or may not have a maximum element. For example, 42 has maximum 41 and ω+6 has maximum ω+5. On the other hand, ω does not have a maximum since there is no largest natural number. If an ordinal has a maximum α, then it is the next ordinal after
754:
natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them as names.) After all of these come ω·2 (which is ω+ω), ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set
846:
It is inappropriate to distinguish between two well-ordered sets if they only differ in the "labeling of their elements", or more formally: if the elements of the first set can be paired off with the elements of the second set such that if one element is smaller than another in the first set, then
2779:
A class is stationary if it has a nonempty intersection with every closed unbounded class. All superclasses of closed unbounded classes are stationary, and stationary classes are unbounded, but there are stationary classes that are not closed and stationary classes that have no closed unbounded
2113:
of ordinals (a collection of ordinals, possibly too large to form a set, defined by some property): any class of ordinals can be indexed by ordinals (and, when the class is unbounded in the class of all ordinals, this puts it in class-bijection with the class of all ordinals). So the
3073:
as the cardinal's representation. (However, we must then be careful to distinguish between cardinal arithmetic and ordinal arithmetic.) In set theories without the axiom of choice, a cardinal may be represented by the set of sets with that cardinality having minimal rank (see
3008:
There are three usual operations on ordinals: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the operation or by using transfinite recursion. The
3798:
An ordinal that is equal to its cofinality is called regular and it is always an initial ordinal. Any limit of regular ordinals is a limit of initial ordinals and thus is also initial even if it is not regular, which it usually is not. If the Axiom of Choice, then
980:, defines the order type of a well-ordering as the set of all well-orderings similar (order-isomorphic) to that well-ordering: in other words, an ordinal number is genuinely an equivalence class of well-ordered sets. This definition must be abandoned in
1114:
of well-ordered sets, it will be defined as a particular well-ordered set that (canonically) represents the class. Thus, an ordinal number will be a well-ordered set; and every well-ordered set will be order-isomorphic to exactly one ordinal number.
5988:
1976:
that is, the smallest ordinal (it always exists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself). Put more directly, it is the supremum of the set of smaller ordinals.
3034:, a game-theoretic variant of numbers, ordinals can also be combined via nimber arithmetic operations. These operations are commutative but the restriction to natural numbers is generally not the same as ordinary addition of natural numbers.
3013:
provides a standardized way of writing ordinals. It uniquely represents each ordinal as a finite sum of ordinal powers of ω. However, this cannot form the basis of a universal ordinal notation due to such self-referential representations as
3068:
is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In theories with the axiom of choice, the cardinal number of any set has an initial ordinal, and one may employ the
3155:. It may be clearer to apply Von Neumann cardinal assignment to finite cases and to use Scott's trick for sets which are infinite or do not admit well orderings. Note that cardinal and ordinal arithmetic agree for finite numbers.
1999:
Thus, every ordinal is either zero, or a successor (of a well-defined predecessor), or a limit. This distinction is important, because many definitions by transfinite recursion rely upon it. Very often, when defining a function
3063:
of that cardinal. Every finite ordinal (natural number) is initial, and no other ordinal associates with its cardinal. But most infinite ordinals are not initial, as many infinite ordinals associate with the same cardinal. The
4728:
having countably many predecessors—that is, the set of countable ordinals—is the union of these two number classes. Cantor proved that the cardinality of the second number class is the first uncountable cardinality.
3059:), then they associate with the same cardinal. Any well-ordered set having an ordinal as its order-type has the same cardinality as that ordinal. The least ordinal associated with a given cardinal is called the
1827:
2032:
nondecreasing and taking ordinal values) are called continuous. Ordinal addition, multiplication and exponentiation are continuous as functions of their second argument (but can be defined non-recursively).
779:) (to give a few examples of relatively small—countable—ordinals). This can be continued indefinitely (as every time one says "and so on" when enumerating ordinals, it defines a larger ordinal). The smallest
2134:-th element in the class (with the convention that the "0-th" is the smallest, the "1-st" is the next smallest, and so on) can be freely spoken of. Formally, the definition is by transfinite induction: the
921:. Provided there exists an order isomorphism between two well-ordered sets, the order isomorphism is unique: this makes it quite justifiable to consider the two sets as essentially identical, and to seek a
5996:
3686:. Notice that a number of authors define cofinality or use it only for limit ordinals. The cofinality of a set of ordinals or any other well-ordered set is the cofinality of the order type of that set.
3436:
is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers: each such well-ordering defines a countable ordinal, and
2730:(to avoid talking of topology on proper classes, one can demand that the intersection of the class with any given ordinal is closed for the order topology on that ordinal, this is again equivalent).
1974:
1891:
1228:
4257:
3025:, and the so-called "natural" arithmetical operations for surreal numbers are an alternative way to combine ordinals arithmetically. They retain commutativity at the expense of continuity.
314:
4370:
4300:
4116:
4171:
4023:
2411:
1756:
3407:
2440:
4200:
2766:
1276:
629:
Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are called
3741:
ranges over the natural numbers) tends to ω; but, more generally, any countable limit ordinal has cofinality ω. An uncountable limit ordinal may have either cofinality ω as does
2277:
3830:
771:
are natural numbers) must itself have an ordinal associated with it: and that is ω. Further on, there will be ω, then ω, and so on, and ω, then ω, then later ω, and even later ε
2720:
1156:
5578:
5492:
5028:
4078:
3941:
3766:
3546:
3216:
3185:
2572:
2336:
2226:
2178:
363:
4403:
1920:
3347:
839:
and there is no infinite decreasing sequence (the latter being easier to visualize). In practice, the importance of well-ordering is justified by the possibility of applying
750:
Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, ... After
6495:
3127:
5529:
5453:
4983:
6810:
6733:
6694:
6656:
6628:
6600:
6572:
6460:
6427:
6399:
6371:
723:). Indeed, the set of natural numbers is well-ordered—as is any set of ordinals—and since it is downward closed, it can be identified with the ordinal associated with it.
5408:
4716:
having countably many predecessors. To define this set, he defined the transfinite ordinal numbers and transformed the infinite indices into ordinals by replacing ∞ with
4327:
3912:
3883:
3604:
3573:
3517:
3488:
3461:
3434:
3374:
3315:
3286:
3251:
1996:ω is a limit ordinal because for any smaller ordinal (in this example, a natural number) there is another ordinal (natural number) larger than it, but still less than ω.
2694:
2645:
498:
468:
7532:
4047:
3852:
3791:
3729:
3707:
3682:
3656:
3636:
2992:
2970:
2945:
2921:
2901:
2881:
2861:
2841:
2821:
2801:
2665:
2616:
2504:
2378:
2358:
2307:
2251:
2152:
2132:
2105:
2081:
2059:
1847:
925:
of the isomorphism type (class). This is exactly what the ordinals provide, and it also provides a canonical labeling of the elements of any well-ordered set. Every
810:
721:
699:
522:
440:
4222:
4138:
2526:
2198:
847:
the partner of the first element is smaller than the partner of the second element in the second set, and vice versa. Such a one-to-one correspondence is called an
3151:
3099:
2596:
2546:
2474:
336:
956:
of "being order-isomorphic". There is a technical difficulty involved, however, in the fact that the equivalence class is too large to be a set in the usual
6111:
2028:
does not take ordinal values). Thus, the interesting step in the definition is the successor step, not the limit ordinals. Such functions (especially for
1009:
4439:
if and only if the ordinal is a countable cardinal, i.e. at most ω. A subset of ω + 1 is open in the order topology if and only if either it is
6251:
6227:
1407:
The class of all ordinals is not a set. If it were a set, one could show that it was an ordinal and thus a member of itself, which would contradict its
1984:
There is an ordinal less than α and whenever ζ is an ordinal less than α, then there exists an ordinal ξ such that ζ < ξ < α.
6321:
4512:
as the intersection of these sets. Then he iterated the derived set operation and intersections to extend his sequence of sets into the infinite:
933:,<) is order-isomorphic to the set of ordinals less than one specific ordinal number under their natural ordering. This canonical set is the
2783:
Rather than formulating these definitions for (proper) classes of ordinals, one can formulate them for sets of ordinals below a given ordinal
1629:(β). It then follows by transfinite induction that there is one and only one function satisfying the recursion formula up to and including α.
5907:
5863:
5792:
5691:
5130:
660:
the set of ordinals that precede it. For example, the ordinal 42 is generally identified as the set {0, 1, 2, ..., 41}. Conversely, any set
7221:
7034:
30:
This article is about the mathematical concept. For number words denoting a position in a sequence ("first", "second", "third", etc.), see
1785:
422:(this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number
7549:
618:, and the notion of position, which leads to the ordinal numbers described here. This is because while any set has only one size (its
419:
6026:
5974:
5945:
5886:
5834:
5811:
5739:
5716:
2285:
247:
229:
207:
167:
70:
2578:
when the limit of a sequence of ordinals in the class is again in the class: or, equivalently, when the indexing (class-)function
1400:, the ordinal obtained by taking the union of all the ordinals in the set. This union exists regardless of the set's size, by the
6104:
1597:
Transfinite induction can be used not only to prove things, but also to define them. Such a definition is normally said to be by
7527:
6152:
2085:. This applies, in particular, to any set of ordinals: any set of ordinals is naturally indexed by the ordinals less than some
981:
957:
7121:
6883:
3070:
1385:. Further, every set of ordinals is well-ordered. This generalizes the fact that every set of natural numbers is well-ordered.
101:
56:
3771:
The cofinality of 0 is 0. And the cofinality of any successor ordinal is 1. The cofinality of any limit ordinal is at least
5601:
4893:
Using successors, limits, and cardinality, Cantor generated an unbounded sequence of ordinal numbers and number classes. The
1929:
1422:
if and only if the opposite order is also well-ordered, which is the case if and only if each of its non-empty subsets has a
703:, which can be identified with the set of natural numbers (so that the ordinal associated with every natural number precedes
148:
105:
120:
7301:
7180:
6844:
6083:
6058:
4780:. Since there are uncountably many of these pairwise disjoint sets, their union is uncountable. This union is a subset of
2445:
776:
7544:
2024:(β) for all β<δ (either in the sense of ordinal limits, as previously explained, or for some other notion of limit if
1856:
6314:
5918:
1330:
that every well-ordered set is order-isomorphic to exactly one of these ordinals, that is, there is an order preserving
922:
7537:
6191:
7175:
7138:
6470:
6097:
6053:
5993:
Acta litterarum ac scientiarum Ragiae
Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum
1632:
Here is an example of definition by transfinite recursion on the ordinals (more will be given later): define function
1557:
Any property that passes from the set of ordinals smaller than a given ordinal α to α itself, is true of all ordinals.
1488:
1173:
127:
5239:, p. 52 attributes the idea to unpublished work of Zermelo in 1916 and several papers by von Neumann the 1920s.
5044:
is a non-limit ordinal. Therefore, the non-limit number classes partition the ordinals into pairwise disjoint sets.
4229:
6073:
6070:
269:
4724:. The second number class is the set of ordinals whose predecessors form a countably infinite set. The set of all
4329:
in the name, this ordinal is countable), which is the smallest ordinal that cannot in any way be represented by a
4262:
2340:. The technique of indexing classes of ordinals is often useful in the context of fixed points: for example, the
7226:
7111:
7099:
7094:
6878:
6834:
6284:
5063:
4833:
4459:
4338:
4268:
4087:
1532:
832:
784:
642:
568:
134:
7687:
7682:
7027:
7001:
6873:
5708:
4887:
4143:
3995:
2383:
1460:
652:
Any ordinal is defined by the set of ordinals that precede it. In fact, the most common definition of ordinals
200:
194:
988:
because these equivalence classes are too large to form a set. However, this definition still can be used in
2776:
cardinals, however, is unbounded but not closed, and any finite set of ordinals is closed but not unbounded.
1729:
730:
A graphical "matchstick" representation of the ordinal ω. Each stick corresponds to an ordinal of the form ω·
7646:
7564:
7439:
7391:
7205:
7128:
6307:
6120:
5731:
3977:
3379:
2418:
1726:, namely the successor of α, written α+1. In the von Neumann definition of ordinals, the successor of α is
960:(ZF) formalization of set theory. But this is not a serious difficulty. The ordinal can be said to be the
679:
This definition of ordinals in terms of sets allows for infinite ordinals. The smallest infinite ordinal is
614:
When dealing with infinite sets, however, one has to distinguish between the notion of size, which leads to
94:
4178:
2744:
1243:
116:
7598:
7479:
7291:
7104:
6961:
6465:
5413:
5040:-th number class consists of ordinals different from those in the preceding number classes if and only if
4751:
2256:
415:
211:
3802:
7514:
7428:
7348:
7328:
7306:
6946:
6782:
6510:
6505:
5606:
5058:
4120:, etc. Many ordinals can be defined in such a manner as fixed points of certain ordinal functions (the
3659:
2699:
1598:
1544:
1327:
1125:
976:
875:
840:
6208:
5556:
5470:
5006:
4056:
3919:
3744:
3524:
3194:
3163:
2551:
2314:
2205:
2157:
1704:
is exactly transfinite induction). It turns out that this example is not very exciting, since provably
341:
4413:
can also be defined above the Church-Kleene ordinal, which are of interest in various parts of logic.
4381:
1896:
442:(omega) to be the least element that is greater than every natural number, along with ordinal numbers
264:
Representation of the ordinal numbers up to ω. One turn of the spiral corresponds to the mapping
7588:
7578:
7412:
7343:
7296:
7236:
7116:
6699:
6432:
6270:
6181:
6171:
5758:
4901:-th number class is the set of ordinals whose predecessors form a set of the same cardinality as the
3320:
1664:; this apparent vicious circle is exactly what definition by transfinite recursion permits. In fact,
1412:
997:
985:
953:
856:
572:
557:
6476:
6048:
3106:
641:
set such that, given two distinct elements, one is less than the other). Equivalently, assuming the
603:
of an element in a sequence. When restricted to finite sets, these two concepts coincide, since all
7583:
7494:
7407:
7402:
7397:
7211:
7153:
7084:
7020:
6910:
6820:
6777:
6759:
6537:
5501:
5425:
4955:
4330:
3317:
is used when writing cardinals, and ω when writing ordinals (this is important since, for example,
1528:
1495:, one has to further make sure that the definition excludes urelements from appearing in ordinals.
1435:
260:
62:
6793:
6716:
6677:
6639:
6611:
6583:
6555:
6443:
6410:
6382:
6354:
7506:
7501:
7286:
7241:
7148:
6815:
6527:
6256:
5671:
5656:
5623:
5386:
4410:
4305:
3983:
3890:
3861:
3582:
3551:
3495:
3466:
3439:
3412:
3352:
3293:
3264:
3229:
3010:
3003:
2280:
1331:
553:
4458:
The transfinite ordinal numbers, which first appeared in 1883, originated in Cantor's work with
3579:(any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the
2670:
2621:
477:
447:
7363:
7200:
7192:
7163:
7133:
7057:
6973:
6936:
6900:
6839:
6825:
6520:
6500:
6161:
6022:
5970:
5941:
5903:
5882:
5859:
5830:
5807:
5788:
5735:
5728:
From
Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Volume 2
5712:
5687:
5679:
5126:
4595:
4436:
4428:
2723:
2061:; in other words, its elements can be indexed in increasing fashion by the ordinals less than
1722:
1159:
949:
945:
848:
608:
596:
4032:
3837:
3776:
3714:
3692:
3667:
3641:
3621:
2977:
2955:
2930:
2906:
2886:
2866:
2846:
2826:
2806:
2786:
2650:
2601:
2489:
2363:
2343:
2292:
2236:
2137:
2117:
2090:
2066:
2044:
1832:
795:
706:
684:
507:
425:
7651:
7641:
7626:
7621:
7489:
7143:
6920:
6895:
6829:
6738:
6704:
6545:
6515:
6437:
6340:
6014:
6010:
5984:
5844:
5767:
5648:
5615:
5172:
4207:
4123:
3075:
2863:
is less than some ordinal in the set. More generally, one can call a subset of any ordinal
2511:
2183:
1980:
Another way of defining a limit ordinal is to say that α is a limit ordinal if and only if:
1434:
There are other modern formulations of the definition of ordinal. For example, assuming the
1231:
1013:
887:
409:
141:
5184:
4720:, the first transfinite ordinal number. Cantor called the set of finite ordinals the first
1311:
The natural numbers are thus ordinals by this definition. For instance, 2 is an element of
7520:
7458:
7276:
7089:
6868:
6772:
6404:
6131:
5959:
5955:
5878:
5665:
English translation: Contributions to the
Founding of the Theory of Transfinite Numbers II
5180:
4333:(this can be made rigorous, of course). Considerably large ordinals can be defined below
3065:
3048:
993:
836:
780:
665:
634:
615:
549:
542:
534:
382:
31:
1527:. An ordinary sequence corresponds to the case α = ω, while a finite α corresponds to a
418:
that include the natural numbers and have the property that every set of ordinals has a
7656:
7453:
7434:
7323:
7280:
7216:
7158:
6915:
6890:
6709:
6577:
6348:
6140:
5934:
5929:
5823:
5750:
5675:
5076:, a generalization of ordinals which includes negative, real, and infinitesimal values.
5073:
5068:
4447:
4432:
4422:
3136:
3084:
3022:
2727:
2581:
2531:
2459:
1776:
1456:
1401:
1337:
Furthermore, the elements of every ordinal are ordinals themselves. Given two ordinals
1289:
584:
401:
321:
2289:(meaning a nonzero ordinal that is not the sum of two strictly smaller ordinals): the
996:
and related systems (where it affords a rather surprising alternative solution to the
7676:
7661:
7631:
7463:
7377:
7372:
6978:
6951:
6860:
5784:
5700:
5660:
5627:
4375:
2574:(then the class must be a proper class, i.e., it cannot be a set). It is said to be
1767:
1553:
set, but it is so important in relation to ordinals that it is worth restating here.
1358:
863:
638:
637:
set in which every non-empty subset has a least element (a totally ordered set is an
623:
412:
5163:
Hallett, Michael (1979), "Towards a theory of mathematical research programmes. I",
5032:. Therefore, the cardinalities of the number classes correspond one-to-one with the
537:
implies that every set can be well-ordered, and given two well-ordered sets, one is
400:
A finite set can be enumerated by successively labeling each element with the least
17:
7611:
7606:
7424:
7353:
7311:
7170:
7067:
6941:
6743:
5597:
5199:
5033:
4670:
4406:
1296:
604:
564:
405:
394:
5781:
Labyrinth of
Thought: A History of Set Theory and Its Role in Mathematical Thought
4535:
The superscripts containing ∞ are just indices defined by the derivation process.
2154:-th element of the class is defined (provided it has already been defined for all
6065:
5897:
5120:
4761:
be countable, and assume there is no such α. This assumption produces two cases.
7636:
7271:
6767:
6549:
4474:
1772:
1474:
1382:
989:
619:
529:
A linear order such that every non-empty subset has a least element is called a
390:
83:
2283:
for the definition of multiplication of ordinals). Similarly, one can consider
7616:
7484:
7387:
7043:
6748:
6605:
5094:
3965:
3615:
2041:
Any well-ordered set is similar (order-isomorphic) to a unique ordinal number
1550:
1419:
1238:: "each ordinal is the well-ordered set of all smaller ordinals". In symbols,
962:
828:
630:
588:
571:, which he had previously introduced in 1872 while studying the uniqueness of
538:
530:
370:
5751:"'What fermented in me for years': Cantor's discovery of transfinite numbers"
4939:. Its cardinality is the limit of the cardinalities of these number classes.
1711:
for all ordinals α, which can be shown, precisely, by transfinite induction.
831:
set, every non-empty subset contains a distinct smallest element. Given the
7419:
7382:
7333:
7231:
1601:– the proof that the result is well-defined uses transfinite induction. Let
1585:
for all ordinals α, one can assume that it is already known for all smaller
1492:
891:
6089:
5771:
1230:
ordered by inclusion. This motivates the standard definition, suggested by
5176:
6856:
6787:
6633:
5053:
4440:
2734:
1524:
1397:
5664:
2233:
This could be applied, for example, to the class of limit ordinals: the
6376:
5848:
5652:
5619:
3081:
One issue with Scott's trick is that it identifies the cardinal number
2974:, i.e. a limit of ordinals in the set is either in the set or equal to
1423:
726:
4913:-th number class is the cardinality immediately following that of the
3051:, its cardinality. If there is a bijection between two ordinals (e.g.
7444:
7266:
6991:
6330:
3030:
1692:(1) makes sense (it is the smallest ordinal not in the singleton set
1415:. The class of all ordinals is variously called "Ord", "ON", or "∞".
974:
The original definition of ordinal numbers, found for example in the
404:
that has not been previously used. To extend this process to various
5640:
6019:
From Frege to Gödel: A Source Book in
Mathematical Logic, 1879–1931
5925:
Also defines ordinal operations in terms of the Cantor Normal Form.
4374:, however, which measure the "proof-theoretic strength" of certain
851:, and the two well-ordered sets are said to be order-isomorphic or
545:
of the other. So ordinal numbers exist and are essentially unique.
365:
as least fixed point, larger ordinal numbers cannot be represented.
7316:
7076:
6080:
1822:{\displaystyle \langle \alpha _{\iota }|\iota <\gamma \rangle }
725:
259:
6299:
4886:. Cantor's work with derived sets and ordinal numbers led to the
3733:. For example, the cofinality of ω is ω, because the sequence ω·
1771:. One justification for this term is that a limit ordinal is the
1609:
to be defined on the ordinals. The idea now is that, in defining
1299:
well-ordered with respect to set membership and every element of
2733:
Of particular importance are those classes of ordinals that are
1392:
is a set having as elements precisely the ordinals smaller than
567:
in 1883 in order to accommodate infinite sequences and classify
7016:
6303:
6093:
6076:
free software for computing with ordinals and ordinal notations
5854:, in Gabbay, Dov M.; Kanamori, Akihiro; Woods, John H. (eds.),
3956:
is a regular ordinal, i.e. the cofinality of the cofinality of
2927:
some ordinal in the set. The subset is said to be closed under
179:
77:
36:
7012:
5825:
Numbers, Sets, and Axioms : the
Apparatus of Mathematics
6482:
6021:(3rd ed.), Harvard University Press, pp. 346–354,
5960:"Frege versus Cantor and Dedekind: On the Concept of Number"
5705:
Georg Cantor: His
Mathematics and Philosophy of the Infinite
5095:"Ordinal Number - Examples and Definition of Ordinal Number"
4493:. In 1880, he pointed out that these sets form the sequence
5821:
Hamilton, A. G. (1982), "6. Ordinal and cardinal numbers",
3259:(all are countable ordinals). So ω can be identified with
3189:, it is always a limit ordinal. Its cardinality is written
2016:(α) is defined, and then, for limit ordinals δ one defines
783:
ordinal is the set of all countable ordinals, expressed as
5641:"Beitrage zur Begrundung der transfiniten Mengenlehre. II"
5221:
5219:
3490:
is the smallest ordinal whose cardinality is greater than
1775:
in a topological sense of all smaller ordinals (under the
1684:(0) is equal to 0 (the smallest ordinal of all). Now that
4875:
being countable. Therefore, there is a countable ordinal
3832:
is regular for each α. In this case, the ordinals 0, 1,
5602:"Ueber unendliche, lineare Punktmannichfaltigkeiten. 5."
4701:
The second theorem requires proving the existence of an
944:
Essentially, an ordinal is intended to be defined as an
5967:
Early
Analytic Philosophy: Frege, Russell, Wittgenstein
5923:(2nd ed.), Warszawa: Państwowe Wydawnictwo Naukowe
1969:{\displaystyle \{\alpha _{\iota }|\iota <\gamma \},}
1581:α. Or, more practically: in order to prove a property
2004:
by transfinite recursion on all ordinals, one defines
1613:(α) for an unspecified ordinal α, one may assume that
6796:
6719:
6680:
6642:
6614:
6586:
6558:
6479:
6446:
6413:
6385:
6357:
5559:
5504:
5473:
5428:
5389:
5009:
4958:
4691:
is the countable union of countable sets. Therefore,
4384:
4341:
4308:
4271:
4232:
4210:
4181:
4146:
4126:
4090:
4059:
4035:
3998:
3922:
3893:
3864:
3840:
3805:
3779:
3747:
3717:
3695:
3670:
3644:
3624:
3585:
3554:
3527:
3498:
3469:
3442:
3415:
3382:
3355:
3323:
3296:
3267:
3232:
3197:
3166:
3139:
3109:
3087:
2980:
2958:
2933:
2909:
2889:
2869:
2849:
2829:
2809:
2789:
2747:
2702:
2673:
2653:
2624:
2604:
2584:
2554:
2534:
2514:
2492:
2462:
2421:
2386:
2366:
2346:
2317:
2295:
2259:
2239:
2208:
2186:
2160:
2140:
2120:
2093:
2069:
2047:
1932:
1899:
1859:
1835:
1788:
1732:
1246:
1176:
1128:
798:
709:
687:
510:
480:
450:
428:
344:
324:
272:
4673:, so they are countable. Proof of first theorem: If
4508:
and he continued the derivation process by defining
2309:-th additively indecomposable ordinal is indexed as
7597:
7560:
7472:
7362:
7250:
7191:
7075:
7050:
6929:
6855:
6757:
6665:
6536:
6338:
5633:
3131:, which in some formulations is the ordinal number
2109:. The same holds, with a slight modification, for
1886:{\displaystyle \alpha _{\iota }<\alpha _{\rho }}
1829:is an ordinal-indexed sequence, indexed by a limit
676:— is (or can be identified with) an ordinal.
408:, ordinal numbers are defined more generally using
108:. Unsourced material may be challenged and removed.
6804:
6727:
6688:
6650:
6622:
6594:
6566:
6489:
6454:
6421:
6393:
6365:
5933:
5822:
5572:
5523:
5486:
5447:
5402:
5022:
4977:
4712:. To prove this, Cantor considered the set of all
4397:
4364:
4321:
4294:
4251:
4216:
4194:
4165:
4132:
4110:
4072:
4041:
4017:
3935:
3906:
3877:
3846:
3824:
3785:
3760:
3723:
3701:
3676:
3650:
3630:
3598:
3567:
3540:
3511:
3482:
3455:
3428:
3401:
3368:
3341:
3309:
3280:
3245:
3210:
3179:
3145:
3121:
3093:
2986:
2964:
2939:
2915:
2895:
2875:
2855:
2835:
2815:
2795:
2760:
2714:
2688:
2659:
2639:
2610:
2590:
2566:
2540:
2520:
2498:
2468:
2434:
2405:
2372:
2352:
2330:
2301:
2271:
2245:
2220:
2192:
2172:
2146:
2126:
2099:
2075:
2053:
1968:
1914:
1885:
1841:
1821:
1750:
1270:
1222:
1150:
804:
715:
693:
516:
492:
462:
434:
357:
330:
308:
5165:The British Journal for the Philosophy of Science
3709:-indexed strictly increasing sequence with limit
1381:, or they are equal. So every set of ordinals is
4739:is countable, then there is a countable ordinal
2722:; this is also the same as being closed, in the
2253:-th ordinal, which is either a limit or zero is
1758:since its elements are those of α and α itself.
1223:{\displaystyle T_{<a}:=\{x\in T\mid x<a\}}
970:Definition of an ordinal as an equivalence class
591:) can be used for two purposes: to describe the
5995:, vol. 1, pp. 199–208, archived from
4409:). Large countable ordinals such as countable
1926:is defined as the least upper bound of the set
1507:is a set, an α-indexed sequence of elements of
870:, and a partial order ≤' is defined on the set
835:, this is equivalent to saying that the set is
6252:the theories of iterated inductive definitions
5647:, vol. 49, no. 2, pp. 207–246,
4252:{\displaystyle \varepsilon _{\alpha }=\alpha }
2667:-th ordinal in the class) is the limit of all
7028:
6315:
6105:
6086:on set theory is an introduction to ordinals.
5858:, Cambridge University Press, pp. 1–71,
3158:The α-th infinite initial ordinal is written
2947:provided it is closed for the order topology
587:(which, in this context, includes the number
309:{\displaystyle f(\alpha )=\omega (1+\alpha )}
8:
6015:"On the introduction of transfinite numbers"
5856:Sets and Extensions in the Twentieth Century
3116:
3110:
2180:), as the smallest element greater than the
1960:
1933:
1816:
1789:
1745:
1739:
1217:
1193:
6033:
5804:Cantorian Set Theory and Limitation of Size
5225:
4365:{\displaystyle \omega _{1}^{\mathrm {CK} }}
4295:{\displaystyle \omega _{1}^{\mathrm {CK} }}
4111:{\displaystyle \omega ^{\omega ^{\omega }}}
4027:, so it is the limit of the sequence 0, 1,
3949:are initial ordinals that are not regular.
2823:is said to be unbounded (or cofinal) under
2769:
1640:(α) be the smallest ordinal not in the set
1396:. For example, every set of ordinals has a
1234:at the age of 19, now called definition of
1010:Set-theoretic definition of natural numbers
71:Learn how and when to remove these messages
7035:
7021:
7013:
6997:
6322:
6308:
6300:
6112:
6098:
6090:
5534:
4587:These theorems are proved by partitioning
4466:is a set of real numbers, the derived set
4204:, then one could go on trying to find the
1668:(0) makes sense since there is no ordinal
1660:(β) known in the very process of defining
1523:, is a generalization of the concept of a
1017:
668:— meaning that for any ordinal α in
27:Generalization of "n-th" to infinite cases
6798:
6797:
6795:
6721:
6720:
6718:
6682:
6681:
6679:
6644:
6643:
6641:
6616:
6615:
6613:
6588:
6587:
6585:
6560:
6559:
6557:
6481:
6480:
6478:
6448:
6447:
6445:
6415:
6414:
6412:
6387:
6386:
6384:
6359:
6358:
6356:
5564:
5558:
5509:
5503:
5478:
5472:
5433:
5427:
5394:
5388:
5333:
5309:
5305:
5293:
5289:
5277:
5265:
5261:
5014:
5008:
4963:
4957:
4905:-th number class. The cardinality of the
4538:Cantor used these sets in the theorems:
4450:section of the "Order topology" article.
4389:
4383:
4352:
4351:
4346:
4340:
4313:
4307:
4282:
4281:
4276:
4270:
4237:
4231:
4209:
4186:
4180:
4166:{\displaystyle \omega ^{\alpha }=\alpha }
4151:
4145:
4125:
4100:
4095:
4089:
4064:
4058:
4034:
4018:{\displaystyle \omega ^{\alpha }=\alpha }
4003:
3997:
3927:
3921:
3898:
3892:
3869:
3863:
3839:
3810:
3804:
3778:
3752:
3746:
3716:
3694:
3689:Thus for a limit ordinal, there exists a
3669:
3643:
3623:
3590:
3584:
3559:
3553:
3532:
3526:
3503:
3497:
3474:
3468:
3447:
3441:
3420:
3414:
3387:
3381:
3360:
3354:
3333:
3328:
3322:
3301:
3295:
3272:
3266:
3255:, which is also the cardinality of ω or ε
3237:
3231:
3202:
3196:
3171:
3165:
3138:
3108:
3086:
2979:
2957:
2932:
2908:
2888:
2868:
2848:
2828:
2808:
2788:
2752:
2746:
2701:
2672:
2652:
2623:
2603:
2583:
2553:
2533:
2513:
2491:
2461:
2426:
2420:
2406:{\displaystyle \omega ^{\alpha }=\alpha }
2391:
2385:
2365:
2345:
2322:
2316:
2294:
2258:
2238:
2207:
2185:
2159:
2139:
2119:
2092:
2068:
2046:
1946:
1940:
1931:
1898:
1877:
1864:
1858:
1834:
1802:
1796:
1787:
1731:
1438:, the following are equivalent for a set
1245:
1181:
1175:
1139:
1127:
797:
708:
686:
626:of any infinite set, as explained below.
509:
479:
449:
427:
349:
343:
323:
271:
248:Learn how and when to remove this message
230:Learn how and when to remove this message
168:Learn how and when to remove this message
5989:"Zur Einführung der transfiniten Zahlen"
5829:, New York: Cambridge University Press,
4443:or it does not contain ω as an element.
3990:is the smallest satisfying the equation
3021:Ordinals are a subclass of the class of
1688:(0) is known, the definition applied to
855:(with the understanding that this is an
556:, although none of these operations are
193:This article includes a list of general
5465:-th number classes, its cardinality is
5381:The first number class has cardinality
5357:
5086:
2772:, are all closed unbounded; the set of
1751:{\displaystyle \alpha \cup \{\alpha \}}
1483:is a transitive set of transitive sets.
672:and any ordinal β < α, β is also in
5345:
5321:
4917:-th number class. For a limit ordinal
4669:have no limit points. Hence, they are
4485:by applying the derived set operation
4427:Any ordinal number can be made into a
3402:{\displaystyle \omega ^{2}>\omega }
2435:{\displaystyle \varepsilon _{\gamma }}
1110:Rather than defining an ordinal as an
897:that preserves the ordering. That is,
755:of ordinals formed in this way (the ω·
5461:-th number class is the union of the
5249:
4925:-th number class is the union of the
4571:is countable, then there is an index
4481:. In 1872, Cantor generated the sets
4195:{\displaystyle \varepsilon _{\iota }}
2761:{\displaystyle \varepsilon _{\cdot }}
2598:is continuous in the sense that, for
1648:, that is, the set consisting of all
1271:{\displaystyle \lambda =[0,\lambda )}
948:of well-ordered sets: that is, as an
7:
5369:
5236:
5150:
5146:
5145:Thorough introductions are given by
4732:Cantor's second theorem becomes: If
3220:. For example, the cardinality of ω
2272:{\displaystyle \omega \cdot \gamma }
1487:These definitions cannot be used in
1411:ordering by membership. This is the
992:and in Quine's axiomatic set theory
554:added, multiplied, and exponentiated
502:, etc., which are even greater than
106:adding citations to reliable sources
6511:Set-theoretically definable numbers
3825:{\displaystyle \omega _{\alpha +1}}
1549:Transfinite induction holds in any
1019:First several von Neumann ordinals
579:Ordinals extend the natural numbers
5561:
5506:
5475:
5430:
5391:
5248:Cantor 1883. English translation:
5011:
4960:
4868:is uncountable, which contradicts
4356:
4353:
4286:
4283:
3964:. So the cofinality operation is
3500:
3357:
3325:
3298:
3269:
3234:
3199:
3113:
2715:{\displaystyle \gamma <\delta }
2286:additively indecomposable ordinals
1151:{\displaystyle a\mapsto T_{<a}}
1004:Von Neumann definition of ordinals
799:
548:Ordinal numbers are distinct from
199:it lacks sufficient corresponding
25:
6228:Takeuti–Feferman–Buchholz ordinal
5849:"Set Theory from Cantor to Cohen"
5686:, Springer, pp. 266–7, 274,
5573:{\displaystyle \aleph _{\alpha }}
5551:-th number class has cardinality
5487:{\displaystyle \aleph _{\omega }}
5420:-th number class has cardinality
5188:. See the footnote on p. 12.
5023:{\displaystyle \aleph _{\alpha }}
5001:-th number class has cardinality
4950:-th number class has cardinality
4073:{\displaystyle \omega ^{\omega }}
3960:is the same as the cofinality of
3936:{\displaystyle \omega _{\omega }}
3761:{\displaystyle \omega _{\omega }}
3541:{\displaystyle \omega _{\omega }}
3211:{\displaystyle \aleph _{\alpha }}
3180:{\displaystyle \omega _{\alpha }}
3047:Each ordinal associates with one
2903:provided every ordinal less than
2567:{\displaystyle \alpha <\beta }
2452:Closed unbounded sets and classes
2331:{\displaystyle \omega ^{\gamma }}
2221:{\displaystyle \beta <\gamma }
2173:{\displaystyle \beta <\gamma }
358:{\displaystyle \omega ^{\omega }}
52:This article has multiple issues.
7066:
6996:
6017:, in Jean van Heijenoort (ed.),
5969:, Open Court, pp. 213–248,
5119:Sterling, Kristin (2007-09-01).
4398:{\displaystyle \varepsilon _{0}}
3463:is the order type of that set),
1915:{\displaystyle \iota <\rho ,}
1104:{∅,{∅},{∅,{∅}},{∅,{∅},{∅,{∅}}}}
622:), there are many nonisomorphic
184:
82:
41:
5726:Ewald, William B., ed. (1996),
4776:is non-empty for all countable
3972:Some "large" countable ordinals
3342:{\displaystyle \aleph _{0}^{2}}
3071:Von Neumann cardinal assignment
2843:provided any ordinal less than
2803:: A subset of a limit ordinal
2773:
1617:(β) is already defined for all
93:needs additional citations for
60:or discuss these issues on the
6490:{\displaystyle {\mathcal {P}}}
4836:, so it is uncountable. Since
3952:The cofinality of any ordinal
3768:or an uncountable cofinality.
3122:{\displaystyle \{\emptyset \}}
2683:
2677:
2634:
2628:
1988:So in the following sequence:
1947:
1803:
1656:. This definition assumes the
1265:
1253:
1166:and the set of all subsets of
1132:
303:
291:
282:
276:
1:
6845:Plane-based geometric algebra
6259: < ω
5524:{\displaystyle \aleph _{n-1}}
5448:{\displaystyle \aleph _{n-1}}
4978:{\displaystyle \aleph _{n-1}}
4655:contains the limit points of
3043:Initial ordinal of a cardinal
1489:non-well-founded set theories
6805:{\displaystyle \mathbb {S} }
6728:{\displaystyle \mathbb {C} }
6689:{\displaystyle \mathbb {R} }
6651:{\displaystyle \mathbb {O} }
6623:{\displaystyle \mathbb {H} }
6595:{\displaystyle \mathbb {C} }
6567:{\displaystyle \mathbb {R} }
6455:{\displaystyle \mathbb {A} }
6422:{\displaystyle \mathbb {Q} }
6394:{\displaystyle \mathbb {Z} }
6366:{\displaystyle \mathbb {N} }
6250:Proof-theoretic ordinals of
5965:, in William W. Tait (ed.),
5920:Cardinal and Ordinal Numbers
4804:is empty for some countable
3658:that is the order type of a
2037:Indexing classes of ordinals
1715:Successor and limit ordinals
1621:and thus give a formula for
1388:Consequently, every ordinal
563:Ordinals were introduced by
6054:Encyclopedia of Mathematics
5806:, Oxford University Press,
5631:. Published separately as:
5403:{\displaystyle \aleph _{0}}
4322:{\displaystyle \omega _{1}}
3914:are regular, whereas 2, 3,
3907:{\displaystyle \omega _{2}}
3878:{\displaystyle \omega _{1}}
3599:{\displaystyle \omega _{n}}
3568:{\displaystyle \omega _{n}}
3512:{\displaystyle \aleph _{1}}
3483:{\displaystyle \omega _{2}}
3456:{\displaystyle \omega _{1}}
3429:{\displaystyle \omega _{1}}
3369:{\displaystyle \aleph _{0}}
3310:{\displaystyle \aleph _{0}}
3290:, except that the notation
3281:{\displaystyle \aleph _{0}}
3246:{\displaystyle \aleph _{0}}
1449:is a (von Neumann) ordinal,
420:least or "smallest" element
7704:
7533:von Neumann–Bernays–Gödel
6273: ≥ ω
5902:(2nd ed.), Springer,
5680:"Cantor's Ordinal Numbers"
4420:
3975:
3001:
2768:ordinals, or the class of
2689:{\displaystyle F(\gamma )}
2640:{\displaystyle F(\delta )}
2476:of ordinals is said to be
1761:A nonzero ordinal that is
1605:denote a (class) function
1542:
1118:For each well-ordered set
1007:
923:"canonical" representative
633:. A well-ordered set is a
389:th, etc.) aimed to extend
29:
7334:One-to-one correspondence
7064:
6987:
6835:Algebra of physical space
6285:First uncountable ordinal
6127:
6032:- English translation of
5802:Hallett, Michael (1986),
5064:First uncountable ordinal
4405:measures the strength of
2484:, when given any ordinal
1319:and so it is a subset of
1091:
1074:
1057:
1040:
1023:
1000:of the largest ordinal).
966:of any set in the class.
833:axiom of dependent choice
643:axiom of dependent choice
493:{\displaystyle \omega +2}
463:{\displaystyle \omega +1}
381:, is a generalization of
6891:Extended complex numbers
6874:Extended natural numbers
6153:Feferman–Schütte ordinal
6121:Large countable ordinals
5783:(2nd revised ed.),
5779:Ferreirós, José (2007),
5749:Ferreirós, José (1995),
5709:Harvard University Press
4888:Cantor-Bendixson theorem
4431:by endowing it with the
3982:As mentioned above (see
3638:is the smallest ordinal
2444:. These are called the "
2020:(δ) as the limit of the
1765:a successor is called a
1521:ordinal-indexed sequence
1511:is a function from α to
1503:If α is any ordinal and
1459:, and set membership is
866:≤ is defined on the set
6192:Bachmann–Howard ordinal
5917:Sierpiński, W. (1965),
5732:Oxford University Press
4929:-th number classes for
4042:{\displaystyle \omega }
3978:Large countable ordinal
3847:{\displaystyle \omega }
3786:{\displaystyle \omega }
3724:{\displaystyle \alpha }
3702:{\displaystyle \delta }
3677:{\displaystyle \alpha }
3651:{\displaystyle \delta }
3631:{\displaystyle \alpha }
2987:{\displaystyle \alpha }
2965:{\displaystyle \alpha }
2940:{\displaystyle \alpha }
2916:{\displaystyle \alpha }
2896:{\displaystyle \alpha }
2876:{\displaystyle \alpha }
2856:{\displaystyle \alpha }
2836:{\displaystyle \alpha }
2816:{\displaystyle \alpha }
2796:{\displaystyle \alpha }
2660:{\displaystyle \delta }
2611:{\displaystyle \delta }
2499:{\displaystyle \alpha }
2373:{\displaystyle \alpha }
2353:{\displaystyle \gamma }
2302:{\displaystyle \gamma }
2246:{\displaystyle \gamma }
2147:{\displaystyle \gamma }
2127:{\displaystyle \gamma }
2100:{\displaystyle \alpha }
2076:{\displaystyle \alpha }
2054:{\displaystyle \alpha }
1842:{\displaystyle \gamma }
1491:. In set theories with
984:and related systems of
805:{\displaystyle \Omega }
716:{\displaystyle \omega }
694:{\displaystyle \omega }
517:{\displaystyle \omega }
435:{\displaystyle \omega }
214:more precise citations.
7292:Constructible universe
7112:Constructibility (V=L)
6947:Transcendental numbers
6806:
6783:Hyperbolic quaternions
6729:
6690:
6652:
6624:
6596:
6568:
6491:
6456:
6423:
6395:
6367:
6132:First infinite ordinal
6066:Ordinals at ProvenMath
5772:10.1006/hmat.1995.1003
5639:Cantor, Georg (1897),
5574:
5525:
5488:
5449:
5414:Mathematical induction
5404:
5024:
4979:
4752:proof by contradiction
4399:
4366:
4323:
4296:
4253:
4224:-th ordinal such that
4218:
4217:{\displaystyle \iota }
4196:
4167:
4140:-th ordinal such that
4134:
4133:{\displaystyle \iota }
4112:
4074:
4043:
4019:
3937:
3908:
3879:
3848:
3826:
3787:
3762:
3725:
3703:
3678:
3652:
3632:
3600:
3569:
3542:
3513:
3484:
3457:
3430:
3403:
3370:
3343:
3311:
3282:
3247:
3212:
3181:
3147:
3123:
3095:
3038:Ordinals and cardinals
2998:Arithmetic of ordinals
2988:
2966:
2941:
2917:
2897:
2877:
2857:
2837:
2817:
2797:
2762:
2716:
2690:
2661:
2641:
2612:
2592:
2568:
2542:
2522:
2521:{\displaystyle \beta }
2500:
2470:
2436:
2407:
2374:
2354:
2332:
2303:
2273:
2247:
2222:
2194:
2193:{\displaystyle \beta }
2174:
2148:
2128:
2101:
2077:
2055:
1970:
1916:
1887:
1843:
1823:
1752:
1720:α, and it is called a
1625:(α) in terms of these
1519:(if α is infinite) or
1272:
1224:
1152:
806:
747:
717:
695:
518:
494:
464:
436:
366:
359:
332:
310:
7515:Principia Mathematica
7349:Transfinite induction
7208:(i.e. set difference)
6879:Extended real numbers
6807:
6730:
6700:Split-complex numbers
6691:
6653:
6625:
6597:
6569:
6492:
6457:
6433:Constructible numbers
6424:
6396:
6368:
5896:Jech, Thomas (2013),
5645:Mathematische Annalen
5607:Mathematische Annalen
5575:
5535:Transfinite induction
5526:
5489:
5450:
5405:
5268:, pp. 159, 204–5
5204:mathworld.wolfram.com
5059:Even and odd ordinals
5025:
4980:
4448:Topology and ordinals
4421:Further information:
4417:Topology and ordinals
4400:
4367:
4324:
4297:
4263:Church–Kleene ordinal
4254:
4219:
4197:
4168:
4135:
4113:
4075:
4044:
4020:
3976:Further information:
3938:
3909:
3880:
3849:
3827:
3788:
3763:
3726:
3704:
3679:
3653:
3633:
3601:
3570:
3543:
3514:
3485:
3458:
3431:
3404:
3371:
3344:
3312:
3283:
3248:
3213:
3182:
3148:
3124:
3096:
2989:
2967:
2942:
2918:
2898:
2878:
2858:
2838:
2818:
2798:
2763:
2717:
2691:
2662:
2642:
2613:
2593:
2569:
2543:
2523:
2501:
2471:
2437:
2408:
2375:
2355:
2333:
2304:
2274:
2248:
2223:
2195:
2175:
2149:
2129:
2102:
2078:
2056:
1971:
1917:
1888:
1844:
1824:
1753:
1599:transfinite recursion
1593:Transfinite recursion
1565:(α) is true whenever
1545:Transfinite induction
1539:Transfinite induction
1328:transfinite induction
1273:
1225:
1153:
977:Principia Mathematica
841:transfinite induction
807:
729:
718:
696:
599:, or to describe the
519:
495:
465:
437:
360:
333:
311:
263:
7589:Burali-Forti paradox
7344:Set-builder notation
7297:Continuum hypothesis
7237:Symmetric difference
6911:Supernatural numbers
6821:Multicomplex numbers
6794:
6778:Dual-complex numbers
6717:
6678:
6640:
6612:
6584:
6556:
6538:Composition algebras
6506:Arithmetical numbers
6477:
6444:
6411:
6383:
6355:
6271:Nonrecursive ordinal
6182:large Veblen ordinal
6172:small Veblen ordinal
5936:Axiomatic Set Theory
5759:Historia Mathematica
5557:
5502:
5471:
5426:
5387:
5372:, p. 5 footnote
5007:
4956:
4382:
4339:
4306:
4269:
4230:
4208:
4179:
4144:
4124:
4088:
4057:
4033:
3996:
3920:
3891:
3862:
3838:
3803:
3777:
3745:
3715:
3693:
3668:
3642:
3622:
3583:
3575:for natural numbers
3552:
3548:is the limit of the
3525:
3496:
3467:
3440:
3413:
3380:
3353:
3321:
3294:
3265:
3230:
3195:
3164:
3137:
3107:
3085:
2978:
2956:
2931:
2907:
2887:
2867:
2847:
2827:
2807:
2787:
2745:
2735:closed and unbounded
2700:
2671:
2651:
2622:
2602:
2582:
2552:
2532:
2512:
2490:
2460:
2419:
2384:
2364:
2344:
2315:
2293:
2257:
2237:
2206:
2200:-th element for all
2184:
2158:
2138:
2118:
2091:
2067:
2045:
1992:0, 1, 2, ..., ω, ω+1
1930:
1897:
1857:
1849:and the sequence is
1833:
1786:
1730:
1569:(β) is true for all
1517:transfinite sequence
1499:Transfinite sequence
1473:is a transitive set
1413:Burali-Forti paradox
1303:is also a subset of
1244:
1236:von Neumann ordinals
1174:
1126:
998:Burali-Forti paradox
986:axiomatic set theory
954:equivalence relation
857:equivalence relation
796:
746:are natural numbers.
707:
685:
664:of ordinals that is
607:of a finite set are
573:trigonometric series
508:
478:
448:
426:
342:
322:
270:
102:improve this article
18:Transfinite sequence
7550:Tarski–Grothendieck
6816:Split-biquaternions
6528:Eisenstein integers
6466:Closed-form numbers
6257:Computable ordinals
5684:The Book of Numbers
5496:, the limit of the
5198:Weisstein, Eric W.
5177:10.1093/bjps/30.1.1
5125:. LernerClassroom.
4435:; this topology is
4411:admissible ordinals
4361:
4331:computable function
4291:
3338:
2737:, sometimes called
1515:. This concept, a
1436:axiom of regularity
1365:. Moreover, either
1326:It can be shown by
1020:
7139:Limitation of size
6974:Profinite integers
6937:Irrational numbers
6802:
6725:
6686:
6648:
6620:
6592:
6564:
6521:Gaussian rationals
6501:Computable numbers
6487:
6452:
6419:
6391:
6363:
6209:Buchholz's ordinal
6071:Ordinal calculator
5940:, D.Van Nostrand,
5873:Levy, A. (2002) ,
5653:10.1007/BF01444205
5620:10.1007/bf01446819
5570:
5521:
5484:
5445:
5400:
5308:, pp. 36–37;
5292:, pp. 35–36;
5264:, pp. 34–35;
5252:, pp. 881–920
5020:
4975:
4395:
4362:
4342:
4319:
4292:
4272:
4249:
4214:
4192:
4163:
4130:
4108:
4070:
4039:
4015:
3984:Cantor normal form
3933:
3904:
3875:
3844:
3822:
3783:
3758:
3721:
3699:
3674:
3648:
3628:
3596:
3565:
3538:
3509:
3480:
3453:
3426:
3399:
3366:
3339:
3324:
3307:
3278:
3243:
3208:
3177:
3143:
3119:
3091:
3011:Cantor normal form
3004:Ordinal arithmetic
2984:
2962:
2937:
2913:
2893:
2873:
2853:
2833:
2813:
2793:
2758:
2712:
2686:
2657:
2637:
2608:
2588:
2564:
2538:
2518:
2496:
2466:
2432:
2403:
2370:
2350:
2328:
2299:
2281:ordinal arithmetic
2269:
2243:
2218:
2190:
2170:
2144:
2124:
2097:
2073:
2051:
1966:
1912:
1883:
1839:
1819:
1748:
1700:), and so on (the
1332:bijective function
1315:and 2 is equal to
1268:
1220:
1148:
1018:
802:
748:
713:
691:
514:
490:
460:
432:
367:
355:
328:
306:
7670:
7669:
7579:Russell's paradox
7528:Zermelo–Fraenkel
7429:Dedekind-infinite
7302:Diagonal argument
7201:Cartesian product
7058:Set (mathematics)
7010:
7009:
6921:Superreal numbers
6901:Levi-Civita field
6896:Hyperreal numbers
6840:Spacetime algebra
6826:Geometric algebra
6739:Bicomplex numbers
6705:Split-quaternions
6546:Division algebras
6516:Gaussian integers
6438:Algebraic numbers
6341:definable numbers
6297:
6296:
6162:Ackermann ordinal
6013:(January 2002) ,
6011:von Neumann, John
5985:von Neumann, John
5909:978-3-662-22400-7
5865:978-0-444-51621-3
5845:Kanamori, Akihiro
5794:978-3-7643-8349-7
5693:978-1-4612-4072-3
5132:978-0-8225-8846-7
4750:. Its proof uses
4596:pairwise disjoint
4429:topological space
3521:, and so on, and
3146:{\displaystyle 1}
3094:{\displaystyle 0}
2618:a limit ordinal,
2591:{\displaystyle F}
2541:{\displaystyle C}
2469:{\displaystyle C}
1723:successor ordinal
1477:by set inclusion,
1430:Other definitions
1377:is an element of
1369:is an element of
1349:is an element of
1313:4 = {0, 1, 2, 3},
1160:order isomorphism
1112:equivalence class
1108:
1107:
950:equivalence class
946:isomorphism class
913:) if and only if
849:order isomorphism
823:Well-ordered sets
331:{\displaystyle f}
258:
257:
250:
240:
239:
232:
178:
177:
170:
152:
75:
16:(Redirected from
7695:
7652:Bertrand Russell
7642:John von Neumann
7627:Abraham Fraenkel
7622:Richard Dedekind
7584:Suslin's problem
7495:Cantor's theorem
7212:De Morgan's laws
7070:
7037:
7030:
7023:
7014:
7000:
6999:
6967:
6957:
6869:Cardinal numbers
6830:Clifford algebra
6811:
6809:
6808:
6803:
6801:
6773:Dual quaternions
6734:
6732:
6731:
6726:
6724:
6695:
6693:
6692:
6687:
6685:
6657:
6655:
6654:
6649:
6647:
6629:
6627:
6626:
6621:
6619:
6601:
6599:
6598:
6593:
6591:
6573:
6571:
6570:
6565:
6563:
6496:
6494:
6493:
6488:
6486:
6485:
6461:
6459:
6458:
6453:
6451:
6428:
6426:
6425:
6420:
6418:
6405:Rational numbers
6400:
6398:
6397:
6392:
6390:
6372:
6370:
6369:
6364:
6362:
6324:
6317:
6310:
6301:
6281:
6280:
6267:
6266:
6114:
6107:
6100:
6091:
6062:
6049:"Ordinal number"
6034:von Neumann 1923
6031:
6006:
6005:
6004:
5979:
5964:
5956:Tait, William W.
5950:
5939:
5924:
5912:
5891:
5875:Basic Set Theory
5868:
5853:
5839:
5828:
5816:
5797:
5774:
5755:
5744:
5721:
5696:
5663:
5630:
5583:
5581:
5579:
5577:
5576:
5571:
5569:
5568:
5546:
5532:
5530:
5528:
5527:
5522:
5520:
5519:
5495:
5493:
5491:
5490:
5485:
5483:
5482:
5456:
5454:
5452:
5451:
5446:
5444:
5443:
5416:proves that the
5411:
5409:
5407:
5406:
5401:
5399:
5398:
5379:
5373:
5367:
5361:
5360:, pp. 61–62
5355:
5349:
5348:, pp. 97–98
5343:
5337:
5336:, pp. 207–8
5331:
5325:
5319:
5313:
5303:
5297:
5287:
5281:
5275:
5269:
5259:
5253:
5246:
5240:
5234:
5228:
5226:von Neumann 1923
5223:
5214:
5213:
5211:
5210:
5200:"Ordinal Number"
5195:
5189:
5187:
5160:
5154:
5143:
5137:
5136:
5116:
5110:
5109:
5107:
5106:
5099:Literary Devices
5091:
5031:
5029:
5027:
5026:
5021:
5019:
5018:
4996:
4986:
4984:
4982:
4981:
4976:
4974:
4973:
4938:
4912:
4900:
4885:
4873:
4866:
4854:
4848:
4846:
4827:
4817:
4803:
4785:
4775:
4759:
4749:
4737:
4711:
4696:
4689:
4679:
4668:
4654:
4648:
4637:
4611:
4604:
4592:
4581:
4569:
4559:
4549:
4534:
4507:
4471:
4407:Peano arithmetic
4404:
4402:
4401:
4396:
4394:
4393:
4373:
4371:
4369:
4368:
4363:
4360:
4359:
4350:
4328:
4326:
4325:
4320:
4318:
4317:
4301:
4299:
4298:
4293:
4290:
4289:
4280:
4260:
4258:
4256:
4255:
4250:
4242:
4241:
4223:
4221:
4220:
4215:
4203:
4201:
4199:
4198:
4193:
4191:
4190:
4172:
4170:
4169:
4164:
4156:
4155:
4139:
4137:
4136:
4131:
4119:
4117:
4115:
4114:
4109:
4107:
4106:
4105:
4104:
4081:
4079:
4077:
4076:
4071:
4069:
4068:
4050:
4048:
4046:
4045:
4040:
4026:
4024:
4022:
4021:
4016:
4008:
4007:
3986:), the ordinal ε
3944:
3942:
3940:
3939:
3934:
3932:
3931:
3913:
3911:
3910:
3905:
3903:
3902:
3886:
3884:
3882:
3881:
3876:
3874:
3873:
3855:
3853:
3851:
3850:
3845:
3831:
3829:
3828:
3823:
3821:
3820:
3794:
3792:
3790:
3789:
3784:
3767:
3765:
3764:
3759:
3757:
3756:
3732:
3730:
3728:
3727:
3722:
3708:
3706:
3705:
3700:
3685:
3683:
3681:
3680:
3675:
3657:
3655:
3654:
3649:
3637:
3635:
3634:
3629:
3605:
3603:
3602:
3597:
3595:
3594:
3574:
3572:
3571:
3566:
3564:
3563:
3547:
3545:
3544:
3539:
3537:
3536:
3520:
3518:
3516:
3515:
3510:
3508:
3507:
3489:
3487:
3486:
3481:
3479:
3478:
3462:
3460:
3459:
3454:
3452:
3451:
3435:
3433:
3432:
3427:
3425:
3424:
3408:
3406:
3405:
3400:
3392:
3391:
3375:
3373:
3372:
3367:
3365:
3364:
3348:
3346:
3345:
3340:
3337:
3332:
3316:
3314:
3313:
3308:
3306:
3305:
3289:
3287:
3285:
3284:
3279:
3277:
3276:
3254:
3252:
3250:
3249:
3244:
3242:
3241:
3219:
3217:
3215:
3214:
3209:
3207:
3206:
3188:
3186:
3184:
3183:
3178:
3176:
3175:
3154:
3152:
3150:
3149:
3144:
3130:
3128:
3126:
3125:
3120:
3100:
3098:
3097:
3092:
3058:
3054:
2993:
2991:
2990:
2985:
2973:
2971:
2969:
2968:
2963:
2946:
2944:
2943:
2938:
2922:
2920:
2919:
2914:
2902:
2900:
2899:
2894:
2882:
2880:
2879:
2874:
2862:
2860:
2859:
2854:
2842:
2840:
2839:
2834:
2822:
2820:
2819:
2814:
2802:
2800:
2799:
2794:
2767:
2765:
2764:
2759:
2757:
2756:
2721:
2719:
2718:
2713:
2695:
2693:
2692:
2687:
2666:
2664:
2663:
2658:
2646:
2644:
2643:
2638:
2617:
2615:
2614:
2609:
2597:
2595:
2594:
2589:
2573:
2571:
2570:
2565:
2547:
2545:
2544:
2539:
2527:
2525:
2524:
2519:
2507:
2505:
2503:
2502:
2497:
2475:
2473:
2472:
2467:
2443:
2441:
2439:
2438:
2433:
2431:
2430:
2412:
2410:
2409:
2404:
2396:
2395:
2379:
2377:
2376:
2371:
2359:
2357:
2356:
2351:
2339:
2337:
2335:
2334:
2329:
2327:
2326:
2308:
2306:
2305:
2300:
2278:
2276:
2275:
2270:
2252:
2250:
2249:
2244:
2229:
2227:
2225:
2224:
2219:
2199:
2197:
2196:
2191:
2179:
2177:
2176:
2171:
2153:
2151:
2150:
2145:
2133:
2131:
2130:
2125:
2108:
2106:
2104:
2103:
2098:
2084:
2082:
2080:
2079:
2074:
2060:
2058:
2057:
2052:
1975:
1973:
1972:
1967:
1950:
1945:
1944:
1921:
1919:
1918:
1913:
1892:
1890:
1889:
1884:
1882:
1881:
1869:
1868:
1848:
1846:
1845:
1840:
1828:
1826:
1825:
1820:
1806:
1801:
1800:
1757:
1755:
1754:
1749:
1710:
1699:
1679:
1671:
1655:
1647:
1620:
1588:
1577:(α) is true for
1572:
1322:
1318:
1314:
1279:
1277:
1275:
1274:
1269:
1232:John von Neumann
1229:
1227:
1226:
1221:
1189:
1188:
1170:having the form
1169:
1165:
1157:
1155:
1154:
1149:
1147:
1146:
1121:
1087:{∅,{∅},{∅,{∅}}}
1021:
1014:Zermelo ordinals
958:Zermelo–Fraenkel
888:order isomorphic
813:
811:
809:
808:
803:
722:
720:
719:
714:
702:
700:
698:
697:
692:
616:cardinal numbers
550:cardinal numbers
525:
523:
521:
520:
515:
501:
499:
497:
496:
491:
471:
469:
467:
466:
461:
441:
439:
438:
433:
410:linearly ordered
388:
385:(first, second,
383:ordinal numerals
364:
362:
361:
356:
354:
353:
337:
335:
334:
329:
317:
315:
313:
312:
307:
253:
246:
235:
228:
224:
221:
215:
210:this article by
201:inline citations
188:
187:
180:
173:
166:
162:
159:
153:
151:
117:"Ordinal number"
110:
86:
78:
67:
45:
44:
37:
21:
7703:
7702:
7698:
7697:
7696:
7694:
7693:
7692:
7688:Wellfoundedness
7683:Ordinal numbers
7673:
7672:
7671:
7666:
7593:
7572:
7556:
7521:New Foundations
7468:
7358:
7277:Cardinal number
7260:
7246:
7187:
7071:
7062:
7046:
7041:
7011:
7006:
6983:
6962:
6952:
6925:
6916:Surreal numbers
6906:Ordinal numbers
6851:
6792:
6791:
6753:
6715:
6714:
6712:
6710:Split-octonions
6676:
6675:
6667:
6661:
6638:
6637:
6610:
6609:
6582:
6581:
6578:Complex numbers
6554:
6553:
6532:
6475:
6474:
6442:
6441:
6409:
6408:
6381:
6380:
6353:
6352:
6349:Natural numbers
6334:
6328:
6298:
6293:
6279:
6276:
6275:
6274:
6265:
6262:
6261:
6260:
6246:
6244:
6223:
6217:
6204:
6158:
6149:
6141:Epsilon numbers
6123:
6118:
6047:
6044:
6039:
6029:
6009:
6002:
6000:
5983:
5977:
5962:
5954:
5948:
5930:Suppes, Patrick
5928:
5916:
5910:
5895:
5889:
5879:Springer-Verlag
5872:
5866:
5851:
5843:
5837:
5820:
5814:
5801:
5795:
5778:
5753:
5748:
5742:
5725:
5719:
5699:
5694:
5672:Conway, John H.
5670:
5638:
5596:
5592:
5587:
5586:
5560:
5555:
5554:
5552:
5538:
5537:proves that if
5505:
5500:
5499:
5497:
5474:
5469:
5468:
5466:
5429:
5424:
5423:
5421:
5390:
5385:
5384:
5382:
5380:
5376:
5368:
5364:
5356:
5352:
5344:
5340:
5332:
5328:
5320:
5316:
5304:
5300:
5288:
5284:
5276:
5272:
5260:
5256:
5247:
5243:
5235:
5231:
5224:
5217:
5208:
5206:
5197:
5196:
5192:
5162:
5161:
5157:
5144:
5140:
5133:
5122:Ordinal Numbers
5118:
5117:
5113:
5104:
5102:
5093:
5092:
5088:
5083:
5050:
5010:
5005:
5004:
5002:
4988:
4959:
4954:
4953:
4951:
4946:is finite, the
4930:
4906:
4894:
4880:
4871:
4864:
4861:In both cases,
4859:
4856:is uncountable.
4852:
4844:
4837:
4819:
4818:, this implies
4809:
4795:
4791:is uncountable.
4783:
4767:
4757:
4744:
4735:
4706:
4694:
4687:
4680:for some index
4674:
4660:
4650:
4639:
4609:
4602:
4599:
4590:
4585:
4576:
4567:
4564:Conversely, if
4557:
4550:for some index
4544:
4513:
4494:
4469:
4456:
4425:
4419:
4385:
4380:
4379:
4337:
4336:
4334:
4309:
4304:
4303:
4267:
4266:
4233:
4228:
4227:
4225:
4206:
4205:
4182:
4177:
4176:
4174:
4147:
4142:
4141:
4122:
4121:
4096:
4091:
4086:
4085:
4083:
4060:
4055:
4054:
4052:
4031:
4030:
4028:
3999:
3994:
3993:
3991:
3989:
3980:
3974:
3948:
3923:
3918:
3917:
3915:
3894:
3889:
3888:
3865:
3860:
3859:
3857:
3836:
3835:
3833:
3806:
3801:
3800:
3775:
3774:
3772:
3748:
3743:
3742:
3713:
3712:
3710:
3691:
3690:
3666:
3665:
3663:
3640:
3639:
3620:
3619:
3612:
3586:
3581:
3580:
3555:
3550:
3549:
3528:
3523:
3522:
3499:
3494:
3493:
3491:
3470:
3465:
3464:
3443:
3438:
3437:
3416:
3411:
3410:
3383:
3378:
3377:
3356:
3351:
3350:
3319:
3318:
3297:
3292:
3291:
3268:
3263:
3262:
3260:
3258:
3233:
3228:
3227:
3225:
3223:
3198:
3193:
3192:
3190:
3167:
3162:
3161:
3159:
3135:
3134:
3132:
3105:
3104:
3102:
3083:
3082:
3066:axiom of choice
3061:initial ordinal
3056:
3052:
3045:
3040:
3028:Interpreted as
3023:surreal numbers
3017:
3006:
3000:
2976:
2975:
2954:
2953:
2951:
2929:
2928:
2905:
2904:
2885:
2884:
2865:
2864:
2845:
2844:
2825:
2824:
2805:
2804:
2785:
2784:
2748:
2743:
2742:
2726:sense, for the
2698:
2697:
2669:
2668:
2649:
2648:
2620:
2619:
2600:
2599:
2580:
2579:
2550:
2549:
2530:
2529:
2510:
2509:
2488:
2487:
2485:
2458:
2457:
2454:
2446:epsilon numbers
2422:
2417:
2416:
2414:
2387:
2382:
2381:
2362:
2361:
2342:
2341:
2318:
2313:
2312:
2310:
2291:
2290:
2255:
2254:
2235:
2234:
2204:
2203:
2201:
2182:
2181:
2156:
2155:
2136:
2135:
2116:
2115:
2089:
2088:
2086:
2065:
2064:
2062:
2043:
2042:
2039:
2012:(α+1) assuming
1936:
1928:
1927:
1895:
1894:
1873:
1860:
1855:
1854:
1831:
1830:
1792:
1784:
1783:
1728:
1727:
1717:
1705:
1693:
1678:(β) | β < 0}
1673:
1669:
1653:
1646:(β) | β < α}
1641:
1618:
1595:
1586:
1570:
1547:
1541:
1501:
1475:totally ordered
1432:
1383:totally ordered
1353:if and only if
1320:
1316:
1312:
1242:
1241:
1239:
1177:
1172:
1171:
1167:
1163:
1135:
1124:
1123:
1119:
1016:
1006:
994:New Foundations
972:
862:Formally, if a
837:totally ordered
825:
820:
794:
793:
791:
788:
774:
705:
704:
683:
682:
680:
666:downward closed
635:totally ordered
581:
543:initial segment
535:axiom of choice
506:
505:
503:
476:
475:
473:
446:
445:
443:
424:
423:
386:
345:
340:
339:
320:
319:
268:
267:
265:
254:
243:
242:
241:
236:
225:
219:
216:
206:Please help to
205:
189:
185:
174:
163:
157:
154:
111:
109:
99:
87:
46:
42:
35:
32:Ordinal numeral
28:
23:
22:
15:
12:
11:
5:
7701:
7699:
7691:
7690:
7685:
7675:
7674:
7668:
7667:
7665:
7664:
7659:
7657:Thoralf Skolem
7654:
7649:
7644:
7639:
7634:
7629:
7624:
7619:
7614:
7609:
7603:
7601:
7595:
7594:
7592:
7591:
7586:
7581:
7575:
7573:
7571:
7570:
7567:
7561:
7558:
7557:
7555:
7554:
7553:
7552:
7547:
7542:
7541:
7540:
7525:
7524:
7523:
7511:
7510:
7509:
7498:
7497:
7492:
7487:
7482:
7476:
7474:
7470:
7469:
7467:
7466:
7461:
7456:
7451:
7442:
7437:
7432:
7422:
7417:
7416:
7415:
7410:
7405:
7395:
7385:
7380:
7375:
7369:
7367:
7360:
7359:
7357:
7356:
7351:
7346:
7341:
7339:Ordinal number
7336:
7331:
7326:
7321:
7320:
7319:
7314:
7304:
7299:
7294:
7289:
7284:
7274:
7269:
7263:
7261:
7259:
7258:
7255:
7251:
7248:
7247:
7245:
7244:
7239:
7234:
7229:
7224:
7219:
7217:Disjoint union
7214:
7209:
7203:
7197:
7195:
7189:
7188:
7186:
7185:
7184:
7183:
7178:
7167:
7166:
7164:Martin's axiom
7161:
7156:
7151:
7146:
7141:
7136:
7131:
7129:Extensionality
7126:
7125:
7124:
7114:
7109:
7108:
7107:
7102:
7097:
7087:
7081:
7079:
7073:
7072:
7065:
7063:
7061:
7060:
7054:
7052:
7048:
7047:
7042:
7040:
7039:
7032:
7025:
7017:
7008:
7007:
7005:
7004:
6994:
6992:Classification
6988:
6985:
6984:
6982:
6981:
6979:Normal numbers
6976:
6971:
6949:
6944:
6939:
6933:
6931:
6927:
6926:
6924:
6923:
6918:
6913:
6908:
6903:
6898:
6893:
6888:
6887:
6886:
6876:
6871:
6865:
6863:
6861:infinitesimals
6853:
6852:
6850:
6849:
6848:
6847:
6842:
6837:
6823:
6818:
6813:
6800:
6785:
6780:
6775:
6770:
6764:
6762:
6755:
6754:
6752:
6751:
6746:
6741:
6736:
6723:
6707:
6702:
6697:
6684:
6671:
6669:
6663:
6662:
6660:
6659:
6646:
6631:
6618:
6603:
6590:
6575:
6562:
6542:
6540:
6534:
6533:
6531:
6530:
6525:
6524:
6523:
6513:
6508:
6503:
6498:
6484:
6468:
6463:
6450:
6435:
6430:
6417:
6402:
6389:
6374:
6361:
6345:
6343:
6336:
6335:
6329:
6327:
6326:
6319:
6312:
6304:
6295:
6294:
6292:
6291:
6282:
6277:
6268:
6263:
6254:
6248:
6240:
6238:
6225:
6219:
6215:
6206:
6202:
6189:
6179:
6169:
6159:
6156:
6150:
6147:
6138:
6128:
6125:
6124:
6119:
6117:
6116:
6109:
6102:
6094:
6088:
6087:
6077:
6068:
6063:
6043:
6042:External links
6040:
6038:
6037:
6027:
6007:
5981:
5975:
5952:
5946:
5926:
5914:
5908:
5893:
5887:
5870:
5864:
5841:
5835:
5818:
5812:
5799:
5793:
5776:
5746:
5740:
5723:
5717:
5701:Dauben, Joseph
5697:
5692:
5668:
5636:
5614:(4): 545–591,
5593:
5591:
5588:
5585:
5584:
5567:
5563:
5518:
5515:
5512:
5508:
5481:
5477:
5442:
5439:
5436:
5432:
5397:
5393:
5374:
5362:
5350:
5338:
5334:Ferreirós 2007
5326:
5314:
5310:Ferreirós 2007
5306:Ferreirós 1995
5298:
5294:Ferreirós 2007
5290:Ferreirós 1995
5282:
5278:Ferreirós 2007
5270:
5266:Ferreirós 2007
5262:Ferreirós 1995
5254:
5241:
5229:
5215:
5190:
5155:
5138:
5131:
5111:
5085:
5084:
5082:
5079:
5078:
5077:
5074:Surreal number
5071:
5066:
5061:
5056:
5049:
5046:
5017:
5013:
4972:
4969:
4966:
4962:
4858:
4857:
4792:
4763:
4698:is countable.
4584:
4583:
4562:
4540:
4473:is the set of
4455:
4452:
4433:order topology
4423:Order topology
4418:
4415:
4392:
4388:
4378:(for example,
4376:formal systems
4358:
4355:
4349:
4345:
4316:
4312:
4288:
4285:
4279:
4275:
4248:
4245:
4240:
4236:
4213:
4189:
4185:
4162:
4159:
4154:
4150:
4129:
4103:
4099:
4094:
4067:
4063:
4038:
4014:
4011:
4006:
4002:
3987:
3973:
3970:
3946:
3930:
3926:
3901:
3897:
3872:
3868:
3843:
3819:
3816:
3813:
3809:
3782:
3755:
3751:
3720:
3698:
3673:
3647:
3627:
3618:of an ordinal
3611:
3608:
3593:
3589:
3562:
3558:
3535:
3531:
3506:
3502:
3477:
3473:
3450:
3446:
3423:
3419:
3398:
3395:
3390:
3386:
3363:
3359:
3336:
3331:
3327:
3304:
3300:
3275:
3271:
3256:
3240:
3236:
3221:
3205:
3201:
3174:
3170:
3142:
3118:
3115:
3112:
3090:
3044:
3041:
3039:
3036:
3015:
3002:Main article:
2999:
2996:
2983:
2961:
2936:
2912:
2892:
2872:
2852:
2832:
2812:
2792:
2755:
2751:
2728:order topology
2711:
2708:
2705:
2685:
2682:
2679:
2676:
2656:
2636:
2633:
2630:
2627:
2607:
2587:
2563:
2560:
2557:
2537:
2517:
2495:
2465:
2453:
2450:
2429:
2425:
2402:
2399:
2394:
2390:
2369:
2349:
2325:
2321:
2298:
2268:
2265:
2262:
2242:
2217:
2214:
2211:
2189:
2169:
2166:
2163:
2143:
2123:
2096:
2072:
2050:
2038:
2035:
1994:
1993:
1986:
1985:
1965:
1962:
1959:
1956:
1953:
1949:
1943:
1939:
1935:
1911:
1908:
1905:
1902:
1880:
1876:
1872:
1867:
1863:
1838:
1818:
1815:
1812:
1809:
1805:
1799:
1795:
1791:
1777:order topology
1747:
1744:
1741:
1738:
1735:
1716:
1713:
1672:, and the set
1594:
1591:
1559:
1558:
1543:Main article:
1540:
1537:
1500:
1497:
1485:
1484:
1478:
1468:
1457:transitive set
1450:
1431:
1428:
1418:An ordinal is
1402:axiom of union
1334:between them.
1309:
1308:
1290:if and only if
1288:is an ordinal
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1187:
1184:
1180:
1145:
1142:
1138:
1134:
1131:
1106:
1105:
1102:
1099:
1096:
1093:
1089:
1088:
1085:
1082:
1079:
1076:
1072:
1071:
1068:
1065:
1062:
1059:
1055:
1054:
1051:
1048:
1045:
1042:
1038:
1037:
1034:
1031:
1028:
1025:
1005:
1002:
971:
968:
890:if there is a
824:
821:
819:
816:
801:
786:
777:epsilon nought
772:
712:
690:
624:well-orderings
585:natural number
580:
577:
513:
489:
486:
483:
459:
456:
453:
431:
402:natural number
375:ordinal number
352:
348:
327:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
256:
255:
238:
237:
192:
190:
183:
176:
175:
90:
88:
81:
76:
50:
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7700:
7689:
7686:
7684:
7681:
7680:
7678:
7663:
7662:Ernst Zermelo
7660:
7658:
7655:
7653:
7650:
7648:
7647:Willard Quine
7645:
7643:
7640:
7638:
7635:
7633:
7630:
7628:
7625:
7623:
7620:
7618:
7615:
7613:
7610:
7608:
7605:
7604:
7602:
7600:
7599:Set theorists
7596:
7590:
7587:
7585:
7582:
7580:
7577:
7576:
7574:
7568:
7566:
7563:
7562:
7559:
7551:
7548:
7546:
7545:Kripke–Platek
7543:
7539:
7536:
7535:
7534:
7531:
7530:
7529:
7526:
7522:
7519:
7518:
7517:
7516:
7512:
7508:
7505:
7504:
7503:
7500:
7499:
7496:
7493:
7491:
7488:
7486:
7483:
7481:
7478:
7477:
7475:
7471:
7465:
7462:
7460:
7457:
7455:
7452:
7450:
7448:
7443:
7441:
7438:
7436:
7433:
7430:
7426:
7423:
7421:
7418:
7414:
7411:
7409:
7406:
7404:
7401:
7400:
7399:
7396:
7393:
7389:
7386:
7384:
7381:
7379:
7376:
7374:
7371:
7370:
7368:
7365:
7361:
7355:
7352:
7350:
7347:
7345:
7342:
7340:
7337:
7335:
7332:
7330:
7327:
7325:
7322:
7318:
7315:
7313:
7310:
7309:
7308:
7305:
7303:
7300:
7298:
7295:
7293:
7290:
7288:
7285:
7282:
7278:
7275:
7273:
7270:
7268:
7265:
7264:
7262:
7256:
7253:
7252:
7249:
7243:
7240:
7238:
7235:
7233:
7230:
7228:
7225:
7223:
7220:
7218:
7215:
7213:
7210:
7207:
7204:
7202:
7199:
7198:
7196:
7194:
7190:
7182:
7181:specification
7179:
7177:
7174:
7173:
7172:
7169:
7168:
7165:
7162:
7160:
7157:
7155:
7152:
7150:
7147:
7145:
7142:
7140:
7137:
7135:
7132:
7130:
7127:
7123:
7120:
7119:
7118:
7115:
7113:
7110:
7106:
7103:
7101:
7098:
7096:
7093:
7092:
7091:
7088:
7086:
7083:
7082:
7080:
7078:
7074:
7069:
7059:
7056:
7055:
7053:
7049:
7045:
7038:
7033:
7031:
7026:
7024:
7019:
7018:
7015:
7003:
6995:
6993:
6990:
6989:
6986:
6980:
6977:
6975:
6972:
6969:
6965:
6959:
6955:
6950:
6948:
6945:
6943:
6942:Fuzzy numbers
6940:
6938:
6935:
6934:
6932:
6928:
6922:
6919:
6917:
6914:
6912:
6909:
6907:
6904:
6902:
6899:
6897:
6894:
6892:
6889:
6885:
6882:
6881:
6880:
6877:
6875:
6872:
6870:
6867:
6866:
6864:
6862:
6858:
6854:
6846:
6843:
6841:
6838:
6836:
6833:
6832:
6831:
6827:
6824:
6822:
6819:
6817:
6814:
6789:
6786:
6784:
6781:
6779:
6776:
6774:
6771:
6769:
6766:
6765:
6763:
6761:
6756:
6750:
6747:
6745:
6744:Biquaternions
6742:
6740:
6737:
6711:
6708:
6706:
6703:
6701:
6698:
6673:
6672:
6670:
6664:
6635:
6632:
6607:
6604:
6579:
6576:
6551:
6547:
6544:
6543:
6541:
6539:
6535:
6529:
6526:
6522:
6519:
6518:
6517:
6514:
6512:
6509:
6507:
6504:
6502:
6499:
6472:
6469:
6467:
6464:
6439:
6436:
6434:
6431:
6406:
6403:
6378:
6375:
6350:
6347:
6346:
6344:
6342:
6337:
6332:
6325:
6320:
6318:
6313:
6311:
6306:
6305:
6302:
6290:
6286:
6283:
6272:
6269:
6258:
6255:
6253:
6249:
6243:
6237:
6233:
6229:
6226:
6222:
6214:
6210:
6207:
6201:
6197:
6193:
6190:
6187:
6183:
6180:
6177:
6173:
6170:
6167:
6163:
6160:
6154:
6151:
6146:
6142:
6139:
6137:
6133:
6130:
6129:
6126:
6122:
6115:
6110:
6108:
6103:
6101:
6096:
6095:
6092:
6085:
6084:lecture notes
6082:
6079:Chapter 4 of
6078:
6075:
6072:
6069:
6067:
6064:
6060:
6056:
6055:
6050:
6046:
6045:
6041:
6035:
6030:
6028:0-674-32449-8
6024:
6020:
6016:
6012:
6008:
5999:on 2014-12-18
5998:
5994:
5990:
5986:
5982:
5978:
5976:0-8126-9344-2
5972:
5968:
5961:
5957:
5953:
5949:
5947:0-486-61630-4
5943:
5938:
5937:
5931:
5927:
5922:
5921:
5915:
5911:
5905:
5901:
5900:
5894:
5890:
5888:0-486-42079-5
5884:
5880:
5876:
5871:
5867:
5861:
5857:
5850:
5846:
5842:
5838:
5836:0-521-24509-5
5832:
5827:
5826:
5819:
5815:
5813:0-19-853283-0
5809:
5805:
5800:
5796:
5790:
5786:
5782:
5777:
5773:
5769:
5765:
5761:
5760:
5752:
5747:
5743:
5741:0-19-850536-1
5737:
5733:
5729:
5724:
5720:
5718:0-674-34871-0
5714:
5710:
5706:
5702:
5698:
5695:
5689:
5685:
5681:
5677:
5673:
5669:
5666:
5662:
5658:
5654:
5650:
5646:
5642:
5637:
5634:
5629:
5625:
5621:
5617:
5613:
5609:
5608:
5603:
5599:
5598:Cantor, Georg
5595:
5594:
5589:
5565:
5550:
5545:
5541:
5536:
5516:
5513:
5510:
5479:
5464:
5460:
5440:
5437:
5434:
5419:
5415:
5395:
5378:
5375:
5371:
5366:
5363:
5359:
5354:
5351:
5347:
5342:
5339:
5335:
5330:
5327:
5324:, p. 111
5323:
5318:
5315:
5312:, p. 271
5311:
5307:
5302:
5299:
5296:, p. 207
5295:
5291:
5286:
5283:
5280:, p. 269
5279:
5274:
5271:
5267:
5263:
5258:
5255:
5251:
5245:
5242:
5238:
5233:
5230:
5227:
5222:
5220:
5216:
5205:
5201:
5194:
5191:
5186:
5182:
5178:
5174:
5170:
5166:
5159:
5156:
5152:
5148:
5142:
5139:
5134:
5128:
5124:
5123:
5115:
5112:
5100:
5096:
5090:
5087:
5080:
5075:
5072:
5070:
5069:Ordinal space
5067:
5065:
5062:
5060:
5057:
5055:
5052:
5051:
5047:
5045:
5043:
5039:
5035:
5034:aleph numbers
5015:
5000:
4995:
4991:
4970:
4967:
4964:
4949:
4945:
4940:
4937:
4933:
4928:
4924:
4920:
4916:
4910:
4904:
4898:
4891:
4889:
4883:
4878:
4874:
4867:
4855:
4847:
4840:
4835:
4831:
4826:
4822:
4816:
4812:
4807:
4802:
4798:
4793:
4790:
4786:
4779:
4774:
4770:
4765:
4764:
4762:
4760:
4753:
4747:
4742:
4738:
4730:
4727:
4723:
4719:
4715:
4709:
4704:
4699:
4697:
4690:
4683:
4677:
4672:
4671:discrete sets
4667:
4663:
4658:
4653:
4646:
4642:
4636:
4632:
4628:
4624:
4620:
4616:
4612:
4605:
4597:
4593:
4579:
4574:
4570:
4563:
4561:is countable;
4560:
4553:
4547:
4542:
4541:
4539:
4536:
4532:
4528:
4524:
4520:
4516:
4511:
4505:
4501:
4497:
4492:
4488:
4484:
4480:
4476:
4472:
4465:
4461:
4453:
4451:
4449:
4444:
4442:
4438:
4434:
4430:
4424:
4416:
4414:
4412:
4408:
4390:
4386:
4377:
4347:
4343:
4332:
4314:
4310:
4302:(despite the
4277:
4273:
4264:
4246:
4243:
4238:
4234:
4211:
4187:
4183:
4160:
4157:
4152:
4148:
4127:
4101:
4097:
4092:
4065:
4061:
4036:
4012:
4009:
4004:
4000:
3985:
3979:
3971:
3969:
3967:
3963:
3959:
3955:
3950:
3928:
3924:
3899:
3895:
3870:
3866:
3841:
3817:
3814:
3811:
3807:
3796:
3780:
3769:
3753:
3749:
3740:
3736:
3718:
3696:
3687:
3671:
3661:
3645:
3625:
3617:
3609:
3607:
3591:
3587:
3578:
3560:
3556:
3533:
3529:
3504:
3475:
3471:
3448:
3444:
3421:
3417:
3396:
3393:
3388:
3384:
3361:
3334:
3329:
3302:
3273:
3238:
3203:
3172:
3168:
3156:
3140:
3088:
3079:
3077:
3076:Scott's trick
3072:
3067:
3062:
3050:
3042:
3037:
3035:
3033:
3032:
3026:
3024:
3019:
3012:
3005:
2997:
2995:
2981:
2959:
2950:
2934:
2926:
2923:is less than
2910:
2890:
2870:
2850:
2830:
2810:
2790:
2781:
2777:
2775:
2771:
2753:
2749:
2740:
2736:
2731:
2729:
2725:
2709:
2706:
2703:
2680:
2674:
2654:
2631:
2625:
2605:
2585:
2577:
2561:
2558:
2555:
2535:
2515:
2508:, there is a
2493:
2483:
2479:
2463:
2451:
2449:
2447:
2427:
2423:
2400:
2397:
2392:
2388:
2367:
2347:
2323:
2319:
2296:
2288:
2287:
2282:
2266:
2263:
2260:
2240:
2231:
2215:
2212:
2209:
2187:
2167:
2164:
2161:
2141:
2121:
2112:
2094:
2070:
2048:
2036:
2034:
2031:
2027:
2023:
2019:
2015:
2011:
2007:
2003:
1997:
1991:
1990:
1989:
1983:
1982:
1981:
1978:
1963:
1957:
1954:
1951:
1941:
1937:
1925:
1909:
1906:
1903:
1900:
1878:
1874:
1870:
1865:
1861:
1852:
1836:
1813:
1810:
1807:
1797:
1793:
1780:
1778:
1774:
1770:
1769:
1768:limit ordinal
1764:
1759:
1742:
1736:
1733:
1725:
1724:
1714:
1712:
1708:
1703:
1697:
1691:
1687:
1683:
1680:is empty. So
1677:
1667:
1663:
1659:
1651:
1645:
1639:
1635:
1630:
1628:
1624:
1616:
1612:
1608:
1604:
1600:
1592:
1590:
1584:
1580:
1576:
1568:
1564:
1556:
1555:
1554:
1552:
1546:
1538:
1536:
1534:
1530:
1526:
1522:
1518:
1514:
1510:
1506:
1498:
1496:
1494:
1490:
1482:
1479:
1476:
1472:
1469:
1466:
1462:
1458:
1454:
1451:
1448:
1445:
1444:
1443:
1441:
1437:
1429:
1427:
1425:
1421:
1416:
1414:
1410:
1405:
1403:
1399:
1395:
1391:
1386:
1384:
1380:
1376:
1372:
1368:
1364:
1360:
1359:proper subset
1356:
1352:
1348:
1344:
1340:
1335:
1333:
1329:
1324:
1306:
1302:
1298:
1294:
1291:
1287:
1283:
1282:
1281:
1262:
1259:
1256:
1250:
1247:
1237:
1233:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1190:
1185:
1182:
1178:
1161:
1143:
1140:
1136:
1129:
1116:
1113:
1103:
1100:
1097:
1094:
1090:
1086:
1083:
1080:
1077:
1073:
1069:
1066:
1063:
1060:
1056:
1052:
1049:
1046:
1043:
1039:
1035:
1032:
1029:
1026:
1022:
1015:
1011:
1003:
1001:
999:
995:
991:
987:
983:
979:
978:
969:
967:
965:
964:
959:
955:
951:
947:
942:
940:
936:
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
893:
889:
885:
881:
877:
873:
869:
865:
864:partial order
860:
858:
854:
850:
844:
842:
838:
834:
830:
822:
817:
815:
789:
782:
778:
770:
766:
762:
758:
753:
745:
741:
737:
733:
728:
724:
710:
688:
677:
675:
671:
667:
663:
659:
656:each ordinal
655:
650:
648:
644:
640:
636:
632:
627:
625:
621:
617:
612:
610:
606:
605:linear orders
602:
598:
594:
590:
586:
578:
576:
574:
570:
566:
561:
559:
555:
551:
546:
544:
540:
536:
532:
527:
511:
487:
484:
481:
457:
454:
451:
429:
421:
417:
414:
411:
407:
406:infinite sets
403:
398:
396:
395:infinite sets
392:
384:
380:
376:
372:
350:
346:
325:
300:
297:
294:
288:
285:
279:
273:
262:
252:
249:
234:
231:
223:
213:
209:
203:
202:
196:
191:
182:
181:
172:
169:
161:
150:
147:
143:
140:
136:
133:
129:
126:
122:
119: –
118:
114:
113:Find sources:
107:
103:
97:
96:
91:This article
89:
85:
80:
79:
74:
72:
65:
64:
59:
58:
53:
48:
39:
38:
33:
19:
7612:Georg Cantor
7607:Paul Bernays
7538:Morse–Kelley
7513:
7446:
7445:Subset
7392:hereditarily
7354:Venn diagram
7338:
7312:ordered pair
7227:Intersection
7171:Axiom schema
6963:
6953:
6905:
6768:Dual numbers
6760:hypercomplex
6550:Real numbers
6288:
6241:
6235:
6231:
6220:
6212:
6199:
6195:
6185:
6175:
6165:
6144:
6135:
6052:
6018:
6001:, retrieved
5997:the original
5992:
5966:
5935:
5919:
5898:
5874:
5855:
5824:
5803:
5780:
5763:
5757:
5727:
5704:
5683:
5676:Guy, Richard
5644:
5632:
5611:
5605:
5548:
5543:
5539:
5462:
5458:
5457:. Since the
5417:
5377:
5365:
5358:Hallett 1986
5353:
5341:
5329:
5317:
5301:
5285:
5273:
5257:
5244:
5232:
5207:. Retrieved
5203:
5193:
5168:
5164:
5158:
5141:
5121:
5114:
5103:. Retrieved
5101:. 2017-05-21
5098:
5089:
5041:
5037:
5036:. Also, the
4998:
4993:
4989:
4947:
4943:
4941:
4935:
4931:
4926:
4922:
4918:
4914:
4908:
4902:
4896:
4892:
4881:
4876:
4869:
4862:
4860:
4850:
4842:
4838:
4829:
4824:
4820:
4814:
4810:
4805:
4800:
4796:
4788:
4781:
4777:
4772:
4768:
4755:
4745:
4740:
4733:
4731:
4725:
4722:number class
4721:
4717:
4713:
4707:
4702:
4700:
4692:
4685:
4681:
4675:
4665:
4661:
4656:
4651:
4644:
4640:
4634:
4630:
4626:
4622:
4618:
4614:
4607:
4600:
4588:
4586:
4577:
4572:
4565:
4555:
4551:
4545:
4537:
4530:
4526:
4522:
4518:
4514:
4509:
4503:
4499:
4495:
4490:
4486:
4482:
4478:
4475:limit points
4467:
4463:
4460:derived sets
4457:
4445:
4426:
3981:
3961:
3957:
3953:
3951:
3797:
3770:
3738:
3734:
3688:
3613:
3576:
3157:
3080:
3060:
3057:ω + 1 > ω
3046:
3029:
3027:
3020:
3007:
2948:
2924:
2782:
2778:
2738:
2732:
2575:
2481:
2477:
2455:
2360:-th ordinal
2284:
2232:
2110:
2040:
2029:
2025:
2021:
2017:
2013:
2009:
2005:
2001:
1998:
1995:
1987:
1979:
1923:
1850:
1781:
1766:
1762:
1760:
1721:
1718:
1706:
1701:
1695:
1689:
1685:
1681:
1675:
1665:
1661:
1657:
1649:
1643:
1637:
1633:
1631:
1626:
1622:
1614:
1610:
1606:
1602:
1596:
1582:
1578:
1574:
1566:
1562:
1561:That is, if
1560:
1551:well-ordered
1548:
1520:
1516:
1512:
1508:
1504:
1502:
1486:
1480:
1470:
1464:
1461:trichotomous
1452:
1446:
1439:
1433:
1417:
1408:
1406:
1393:
1389:
1387:
1378:
1374:
1370:
1366:
1362:
1354:
1350:
1346:
1342:
1338:
1336:
1325:
1321:{0, 1, 2, 3}
1310:
1304:
1300:
1292:
1285:
1280:. Formally:
1235:
1117:
1111:
1109:
975:
973:
961:
943:
938:
934:
930:
927:well-ordered
926:
918:
914:
910:
906:
902:
898:
894:
883:
879:
871:
867:
861:
852:
845:
829:well-ordered
826:
768:
764:
760:
756:
751:
749:
743:
739:
735:
731:
678:
673:
669:
661:
657:
653:
651:
649:of the set.
646:
631:well-ordered
628:
613:
600:
592:
582:
569:derived sets
565:Georg Cantor
562:
547:
528:
413:greek letter
399:
378:
374:
368:
244:
226:
217:
198:
164:
155:
145:
138:
131:
124:
112:
100:Please help
95:verification
92:
68:
61:
55:
54:Please help
51:
7637:Thomas Jech
7480:Alternative
7459:Uncountable
7413:Ultrafilter
7272:Cardinality
7176:replacement
7117:Determinacy
6930:Other types
6749:Bioctonions
6606:Quaternions
5346:Dauben 1979
5322:Dauben 1979
5171:(1): 1–25,
4834:perfect set
4659:, the sets
4625:) ∪ ··· ∪ (
2925:or equal to
2883:cofinal in
2724:topological
2413:is written
1636:by letting
1158:defines an
990:type theory
874:, then the
818:Definitions
781:uncountable
620:cardinality
558:commutative
391:enumeration
220:August 2022
212:introducing
158:August 2022
7677:Categories
7632:Kurt Gödel
7617:Paul Cohen
7454:Transitive
7222:Identities
7206:Complement
7193:Operations
7154:Regularity
7122:projective
7085:Adjunction
7044:Set theory
6884:Projective
6857:Infinities
6081:Don Monk's
6003:2013-09-15
5899:Set Theory
5785:Birkhäuser
5590:References
5250:Ewald 1996
5209:2020-08-12
5105:2021-08-31
4879:such that
4849:, the set
4743:such that
4705:such that
4633:) ∪ ··· ∪
4575:such that
4173:is called
3966:idempotent
3662:subset of
3616:cofinality
3610:Cofinality
3409:). Also,
2548:such that
2380:such that
1851:increasing
1698:(0)} = {0}
1493:urelements
1098:{0,1,2,3}
1008:See also:
963:order type
935:order type
654:identifies
647:order type
609:isomorphic
539:isomorphic
531:well-order
371:set theory
195:references
128:newspapers
57:improve it
7565:Paradoxes
7485:Axiomatic
7464:Universal
7440:Singleton
7435:Recursive
7378:Countable
7373:Amorphous
7232:Power set
7149:Power set
7100:dependent
7095:countable
6968:solenoids
6788:Sedenions
6634:Octonions
6059:EMS Press
5766:: 33–42,
5678:(2012) ,
5661:121665994
5628:121930608
5566:α
5562:ℵ
5514:−
5507:ℵ
5480:ω
5476:ℵ
5438:−
5431:ℵ
5392:ℵ
5370:Tait 1997
5237:Levy 1979
5151:Jech 2003
5147:Levy 1979
5016:α
5012:ℵ
4968:−
4961:ℵ
4489:times to
4387:ε
4344:ω
4311:ω
4274:ω
4247:α
4239:α
4235:ε
4212:ι
4188:ι
4184:ε
4161:α
4153:α
4149:ω
4128:ι
4102:ω
4098:ω
4093:ω
4066:ω
4062:ω
4037:ω
4013:α
4005:α
4001:ω
3929:ω
3925:ω
3896:ω
3867:ω
3842:ω
3812:α
3808:ω
3781:ω
3754:ω
3750:ω
3719:α
3697:δ
3672:α
3646:δ
3626:α
3588:ω
3557:ω
3534:ω
3530:ω
3501:ℵ
3472:ω
3445:ω
3418:ω
3397:ω
3385:ω
3358:ℵ
3326:ℵ
3299:ℵ
3270:ℵ
3235:ℵ
3204:α
3200:ℵ
3173:α
3169:ω
3114:∅
3053:ω = 1 + ω
2982:α
2960:α
2935:α
2911:α
2891:α
2871:α
2851:α
2831:α
2811:α
2791:α
2770:cardinals
2754:⋅
2750:ε
2710:δ
2704:γ
2681:γ
2655:δ
2632:δ
2606:δ
2562:β
2556:α
2516:β
2494:α
2478:unbounded
2428:γ
2424:ε
2401:α
2393:α
2389:ω
2368:α
2348:γ
2324:γ
2320:ω
2297:γ
2267:γ
2264:⋅
2261:ω
2241:γ
2216:γ
2210:β
2188:β
2168:γ
2162:β
2142:γ
2122:γ
2095:α
2071:α
2049:α
2008:(0), and
1958:γ
1952:ι
1942:ι
1938:α
1907:ρ
1901:ι
1893:whenever
1879:ρ
1875:α
1866:ι
1862:α
1837:γ
1817:⟩
1814:γ
1808:ι
1798:ι
1794:α
1790:⟨
1743:α
1737:∪
1734:α
1702:and so on
1531:, a.k.a.
1263:λ
1248:λ
1206:∣
1200:∈
1133:↦
892:bijection
886:,≤') are
882:,≤) and (
800:Ω
711:ω
689:ω
512:ω
482:ω
452:ω
430:ω
416:variables
351:ω
347:ω
301:α
289:ω
280:α
63:talk page
7569:Problems
7473:Theories
7449:Superset
7425:Infinite
7254:Concepts
7134:Infinity
7051:Overview
6377:Integers
6339:Sets of
5987:(1923),
5958:(1997),
5932:(1960),
5847:(2012),
5703:(1979),
5600:(1883),
5054:Counting
5048:See also
4828:. Thus,
4808:. Since
4794:Case 2:
4766:Case 1:
4529:⊇ ··· ⊇
4525:⊇ ··· ⊇
4498:⊇ ··· ⊇
4446:See the
4441:cofinite
4437:discrete
3376:whereas
3049:cardinal
2994:itself.
2456:A class
1670:β < 0
1654:β < α
1652:(β) for
1619:β < α
1587:β < α
1571:β < α
1525:sequence
1398:supremum
1297:strictly
1162:between
1081:{0,1,2}
1070:{∅,{∅}}
952:for the
941:,<).
763:, where
601:position
318:. Since
7507:General
7502:Zermelo
7408:subbase
7390: (
7329:Forcing
7307:Element
7279: (
7257:Methods
7144:Pairing
6958:numbers
6790: (
6636: (
6608: (
6580: (
6552: (
6473: (
6471:Periods
6440: (
6407: (
6379: (
6351: (
6333:systems
6155: Γ
6061:, 2001
5580:
5553:
5531:
5498:
5494:
5467:
5455:
5422:
5410:
5383:
5185:0532548
5030:
5003:
4985:
4952:
4684:, then
4554:, then
4454:History
4372:
4335:
4259:
4226:
4202:
4175:
4118:
4084:
4080:
4053:
4049:
4029:
4025:
3992:
3945:, and ω
3943:
3916:
3885:
3858:
3854:
3834:
3793:
3773:
3737:(where
3731:
3711:
3684:
3664:
3660:cofinal
3519:
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3288:
3261:
3253:
3226:
3224:= ω is
3218:
3191:
3187:
3160:
3153:
3133:
3129:
3103:
3031:nimbers
2972:
2952:
2774:regular
2506:
2486:
2482:cofinal
2442:
2415:
2338:
2311:
2228:
2202:
2111:classes
2107:
2087:
2083:
2063:
1853:, i.e.
1709:(α) = α
1573:, then
1424:maximum
1278:
1240:
853:similar
812:
792:
701:
681:
639:ordered
524:
504:
500:
474:
470:
444:
379:ordinal
316:
266:
208:improve
142:scholar
7398:Filter
7388:Finite
7324:Family
7267:Almost
7105:global
7090:Choice
7077:Axioms
6758:Other
6331:Number
6287:
6230:
6211:
6194:
6184:
6174:
6164:
6143:
6134:
6025:
5973:
5944:
5906:
5885:
5862:
5833:
5810:
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5738:
5715:
5690:
5659:
5626:
5547:, the
5183:
5129:
4997:, the
4921:, the
4754:. Let
4649:since
4638:. For
4598:sets:
4533:⊇ ···.
4506:⊇ ···,
3887:, and
2576:closed
1533:string
1420:finite
1409:strict
1317:{0, 1}
1284:A set
1064:{0,1}
876:posets
738:where
541:to an
533:. The
197:, but
144:
137:
130:
123:
115:
7490:Naive
7420:Fuzzy
7383:Empty
7366:types
7317:tuple
7287:Class
7281:large
7242:Union
7159:Union
6966:-adic
6956:-adic
6713:Over
6674:Over
6668:types
6666:Split
6074:GPL'd
5963:(PDF)
5852:(PDF)
5754:(PDF)
5657:S2CID
5624:S2CID
5081:Notes
4987:. If
4934:<
4832:is a
4787:, so
4643:<
4617:) ∪ (
4594:into
4462:. If
3101:with
3018:= ω.
2739:clubs
2647:(the
2480:, or
2279:(see
1924:limit
1782:When
1773:limit
1529:tuple
1455:is a
1373:, or
1357:is a
929:set (
905:) ≤'
827:In a
595:of a
377:, or
373:, an
149:JSTOR
135:books
7403:base
7002:List
6859:and
6023:ISBN
5971:ISBN
5942:ISBN
5904:ISBN
5883:ISBN
5860:ISBN
5831:ISBN
5808:ISBN
5789:ISBN
5736:ISBN
5713:ISBN
5688:ISBN
5149:and
5127:ISBN
4911:+ 1)
4899:+ 1)
3614:The
3394:>
3055:and
2707:<
2696:for
2559:<
2213:<
2165:<
1955:<
1922:its
1904:<
1871:<
1811:<
1341:and
1212:<
1183:<
1141:<
1053:{∅}
1047:{0}
1012:and
937:of (
767:and
742:and
593:size
338:has
121:news
7364:Set
6203:Ω+1
6188:(Ω)
6178:(Ω)
6168:(Ω)
5768:doi
5649:doi
5616:doi
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4942:If
4884:= ∅
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4748:= ∅
4710:= ∅
4678:= ∅
4606:= (
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4548:= ∅
4543:If
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1779:).
1763:not
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369:In
104:by
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5542:≥
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5412:.
5218:^
5202:.
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5097:.
4992:≥
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4813:⊆
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1426:.
1404:.
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814:.
658:as
611:.
583:A
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560:.
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5042:α
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4932:β
4927:β
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4919:α
4915:α
4909:α
4907:(
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4882:P
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4158:=
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3739:m
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2030:F
2026:F
2022:F
2018:F
2014:F
2010:F
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2002:F
1964:,
1961:}
1948:|
1934:{
1910:,
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1746:}
1740:{
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1471:x
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1465:x
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1101:=
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